1、Chapter 7 learning objectives,Learn the closed economy Solow model See how a countrys standard of living depends on its saving and population growth rates Learn how to use the “Golden Rule” to find the optimal savings rate and capital stock,The importance of economic growth,for poor countries,select

2、ed poverty statistics,In the poorest one-fifth of all countries, daily caloric intake is 1/3 lower than in the richest fifth the infant mortality（死亡數(shù)） rate is 200 per 1000 births, compared to 4 per 1000 births in the richest fifth.,selected poverty statistics,In Pakistan, 85% of people live on less

3、than $2/day One-fourth of the poorest countries have had famines（饑荒） during the past 3 decades. (none of the richest countries had famines) Poverty is associated with the oppression of women and minorities,Estimated effects of economic growth,A 10% increase in income is associated with a 6% decrease

4、 in infant mortality Income growth also reduces poverty. Example:,+65%,-12%,1997-99,-25%,+76%,Income and poverty in the world selected countries, 2000,The importance of economic growth,for poor countries,for rich countries,Huge effects from tiny（極小的） differences,In rich countries like the U.S., if g

5、overnment policies or “shocks” have even a small impact on the long-run growth rate, they will have a huge impact on our standard of living in the long run,Huge effects from tiny differences,1,081.4%,243.7%,85.4%,624.5%,169.2%,64.0%,2.5%,2.0%,100 years,50 years,25 years,Huge effects from tiny differ

6、ences,If the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher during the 1990s, the U.S. would have generated an additional $449 billion of income during that decade,The lessons of growth theory,can make a positive difference in the lives of hundreds of mi

7、llions 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 governments policies,The Solow Model,due to Robert Solow, won Nobel Prize（1987） for contributions to the study of economi

8、c growth a major paradigm（范例，示例）: widely used in policy making benchmark against which most recent growth theories are compared looks at the determinants of economic growth and the standard of living in the long run,How Solow model is different from Chapter 3s model,1. K is no longer fixed: investme

9、nt 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.,How Solow model is different from Chapter 3s model,4. No G or T (only to simplify presentation; we can still do fiscal policy experiments) 5.

10、Cosmetic（化妝品） differences.,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),The pr

11、oduction function,Note: this production function exhibits diminishing MPK.,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,The consumption function,s = the saving rate, the fraction of income that is saved (s is an exogenous paramet

12、er) Note: s is the only lowercase variable that is not equal to its uppercase version divided by L Consumption function: c = (1s)y (per worker),Saving and investment,saving (per worker) = y c = y (1s)y = sy National income identity is y = c + i Rearrange to get: i = y c = sy (investment = saving, li

13、ke in chap. 3!) Using the results above, i = sy = sf(k),Output, consumption, and investment,Depreciation, = the rate of depreciation = the fraction of the capital stock that wears out each period,Capital accumulation,The basic idea: Investment makes the capital stock bigger, depreciation makes it sm

14、aller.,Capital accumulation,Change in capital stock = investment depreciation k = i k Since i = sf(k) , this becomes:,k = s f(k) k,The equation of motion for k,the Solow models central equation Determines behavior of capital over time which, in turn, determines behavior of all of the other endogenou

15、s variables because they all depend on k. E.g., income per person: y = f(k) consump. per person: c = (1s) f(k),k = s f(k) k,The steady state,If investment is just enough to cover depreciation sf(k) = k , then capital per worker will remain constant: k = 0. This constant value, denoted k*, is called

16、the steady state capital stock.,k = s f(k) k,The steady state,Moving toward the steady state,k = sf(k) k,Moving toward the steady state,k = sf(k) k,Moving toward the steady state,k = sf(k) k,k2,Moving toward the steady state,k = sf(k) k,k2,Moving toward the steady state,k = sf(k) k,Moving toward the

17、 steady state,k = sf(k) k,k2,k3,Moving toward the steady state,k = sf(k) k,k3,Summary: As long as k k*, investment will exceed depreciation, and k will continue to grow toward k*.,Now you try:,Draw the Solow model diagram, labeling the steady state k*. On the horizontal axis, pick a value greater th

18、an k* for the economys initial capital stock. Label it k1. Show what happens to k over time. Does k move toward the steady state or away from it?,A numerical example,Production function (aggregate):,To derive the per-worker production function, divide through by L:,Then substitute y = Y/L and k = K/

19、L to get,A numerical example, cont.,Assume: s = 0.3 = 0.1 initial value of k = 4.0,Approaching the Steady State: A Numerical Example,Year k y c i k k 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395 2.096 1.467 0.629 0.440 0.189,Approaching the Steady State: A Nume

20、rical Example,Year k y c i k k 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395 2.096 1.467 0.629 0.440 0.189 4 4.584 2.141 1.499 0.642 0.458 0.184 10 5.602 2.367 1.657 0.710 0.560 0.150 25 7.351 2.706 1.894 0.812 0.732 0.080 100 8.962 2.994 2.096 0.898 0.896 0.002

21、 9.000 3.000 2.100 0.900 0.900 0.000,Exercise: solve for the steady state,Continue to assume s = 0.3, = 0.1, and y = k 1/2,Use the equation of motion k = s f(k) k to solve for the steady-state values of k, y, and c.,Solution to exercise:,An increase in the saving rate,An increase in the saving rate

22、raises investment,causing the capital stock to grow toward a new steady state:,Prediction:,Higher s higher k*. And since y = f(k) , higher k* higher y* . Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker

23、in the long run.,International Evidence on Investment Rates and Income per Person,The Golden Rule: introduction,Different values of s lead to different steady states. How do we know which is the “best” steady state? Economic well-being depends on consumption, so the “best” steady state has the highe

24、st possible value of consumption per person: c* = (1s) f(k*) An increase in s leads to higher k* and y*, which may raise c* reduces consumptions share of income (1s), which may lower c* So, how do we find the s and k* that maximize c* ?,The Golden Rule Capital Stock,the Golden Rule level of capital,

25、 the steady state value of k that maximizes consumption.,To find it, first express c* in terms of k*: c* = y* i* = f (k*) i* = f (k*) k*,In general: i = k + k In the steady state: i* = k* because k = 0.,Then, graph f(k*) and k*, and look for the point where the gap between them is biggest.,The Golde

26、n Rule Capital Stock,The Golden Rule Capital Stock,c* = f(k*) k* is biggest where the slope of the production func. equals the slope of the depreciation line:,steady-state capital per worker, k*,MPK = ,The transition to the Golden Rule Steady State,The economy does NOT have a tendency to move toward

27、 the Golden Rule steady state. Achieving the Golden Rule requires that policymakers adjust s. This adjustment leads to a new steady state with higher consumption. But what happens to consumption during the transition to the Golden Rule?,Starting with too much capital,then increasing c* requires a fa

28、ll in s. In the transition to the Golden Rule, consumption is higher at all points in time.,t0,c,i,y,Starting with too little capital,then increasing c* requires an increase in s. Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption.,time,t0,c,i

29、,y,Population Growth,Assume that the population-and labor force- grow at rate n. (n is exogenous),EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02). Then L = n L = 0.02 1000 = 20, so L = 1020 in year 2.,Break-even investment,( + n)k = break-even（得失相當(dāng)?shù)模?investment, th

30、e amount of investment necessary to keep k constant. Break-even investment includes: k to replace capital as it wears out n k to equip new workers with capital (otherwise, k would fall as the existing capital stock would be spread more thinly over a larger population of workers),The equation of moti

31、on for k,With population growth, the equation of motion for k is k = s f(k) ( + n) k,The Solow Model diagram,k = s f(k) ( +n)k,The impact of population growth,Investment, break-even investment,Capital per worker, k,( +n1) k,k1*,An increase in n causes an increase in break-even investment,leading to a lower steady-state level of k.,Prediction:,Higher n lower k*. And since y = f(k) , lower k* lower y* . Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.,I