1、Chapter Seven1CHAPTER 7Economic Growth IIA PowerPointTutorialTo Accompany MACROECONOMICS, 6th. ed.N. Gregory MankiwByMannig J. SimidianChapter Seven2The Solow Growth Model is designed to show howgrowth in the capital stock, growth in the labor force, and advances in technology interact in an economy
2、, and how they affect a nations total output of goods and services.Lets now examine how the model treats the accumulation of capital.Chapter Seven3Chapter Seven4The production function represents the transformation of inputs (labor (L), capital (K), production technology) into outputs (final goods a
3、nd services for a certain time period).The algebraic representation is: Y = F ( K , L )The Production FunctionLets analyze the supply and demand for goods, andsee how much output is produced at any given timeand how this output is allocated among alternative uses. Key Assumption: The Production Func
4、tion has constant returns to scale.zzzChapter Seven5This assumption lets us analyze all quantities relative to the size of the labor force. Set z = 1/L.Y/ L = F ( K / L , 1 )Constant returns to scale imply that the size of the economy asmeasured by the number of workers does not affect the relations
5、hipbetween output per worker and capital per worker. So, from now on,lets denote all quantities in per worker terms in lower case letters.Here is our production function: , where f(k) = F(k,1).y = f( k )Chapter Seven6MPK = f(k k + 1) f (k k)f(k)The production function shows how the amount of capital
6、 perworker k determines the amountof output per worker y = f(k).The slope of the production function is the marginal product of capital: if k increases by 1 unit, y increases by MPK units.1MPKChapter Seven7y = c + i1)2)3)4)Investment = savings. The rate of saving s sis the fraction of output devoted
7、 to investment.Chapter Seven8Here are two forces that influence the capital stock: Investment: expenditure on plant and equipment. Depreciation: wearing out of old capital; causes capital stock to fall.Recall investment per worker i = s s y.Lets substitute the production function for y, we can expre
8、ss investmentper worker as a function of the capital stock per worker:i = s s f(k k)This equation relates the existing stock of capital k to the accumulationof new capital i.Chapter Seven9Investment, s f(k)Output, f (k)c (per worker)i (per worker)y (per worker)The saving rate s determines the alloca
9、tion of output between consumption and investment. For any level of k, output is f(k), investment is s f(k), and consumption is f(k) sf(k).Chapter Seven10Impact of investment and depreciation on the capital stock: D Dk k = i d dkChange incapital stockInvestmentDepreciationRemember investment equals
10、savings so, it can be written:D Dk = s s f(k k) d dkd dk kd dkDepreciation is therefore proportionalto the capital stock.Chapter Seven11Investment and depreciationCapital per worker, k ki* = d dk k*k k*k k1k k2At k k*, investment equals depreciation and capital will not change over time.Below k*, in
11、vestment exceeds depreciation, so the capital stock grows. Investment, s s f(k k)Depreciation, d dk kAbove k*, depreciation exceeds investment, so the capital stock shrinks. Chapter Seven12Investmentand depreciationCapital per worker, k ki* = d dk k*k k1 1*k k2*Depreciation, d dk kInvestment, s s1 1
12、 f(k k)Investment, s s2f(k k)The Solow Model shows that if the saving rate is high, the economy will have a large capital stock and high level of output. If the savingrate is low, the economy will have a small capital stock and alow level of output.An increase inthe saving ratecauses the capitalstoc
13、k to grow to a new steady state.Chapter Seven13The steady-state value of k that maximizes consumption is calledthe Golden Rule Level of Capital. To find the steady-state consumption per worker, we begin with the national income accounts identity: and rearrange it as:c = y - i.This equation holds tha
14、t consumption is output minus investment. Because we want to find steady-state consumption, we substitute steady-state values for output and investment. Steady-state output per worker is f (k*) where k* is the steady-state capital stock per worker. Furthermore, because the capital stock is not chang
15、ing in the steady state, investment is equal to depreciation dk*. Substituting f (k*) for y and dk* for i, we can write steady-state consumption per worker as c* = f (k*) - dk*.y - c + iChapter Seven14c*= f (k*) - dk*.According to this equation, steady-state consumption is whats leftof steady-state
16、output after paying for steady-state depreciation. Itfurther shows that an increase in steady-state capital has two opposing effects on steady-state consumption. On the one hand, more capital means more output. On the other hand, more capital also means that more output must be used to replace capit
17、al that is wearing out.The economys output is used for consumption or investment. In the steady state, investment equals depreciation. Therefore, steady-state consumption is the difference between output f (k*) and depreciation dk*. Steady-state consumption is maximized at the Golden Rule steady sta
18、te. The Golden Rule capital stock is denoted k*gold, and the Golden Ruleconsumption is c*gold. d dk kd dkOutput, f(k)c *goldk*goldChapter Seven15Lets now derive a simple condition that characterizes the Golden Rulelevel of capital. Recall that the slope of the production function is themarginal prod
19、uct of capital MPK. The slope of the dk* line is d.Because these two slopes are equal at k*gold, the Golden Rule canbe described by the equation: MPK = d.At the Golden Rule level of capital, the marginal product of capitalequals the depreciation rate.Keep in mind that the economy does not automatica
20、lly gravitate towardthe Golden Rule steady state. If we want a particular steady-state capitalstock, such as the Golden Rule, we need a particular saving rate tosupport it. Chapter Seven16The basic Solow model shows that capital accumulation, alone,cannot explain sustained economic growth: high rate
21、s of savinglead to high growth temporarily, but the economy eventuallyapproaches a steady state in which capital and output are constant.To explain the sustained economic growth, we must expand theSolow model to incorporate the other two sources of economicgrowth.So, lets add population growth to th
22、e model. Well assume that thepopulation and labor force grow at a constant rate n.Chapter Seven17Like depreciation, population growth is one reason why the capitalstock per worker shrinks. If n is the rate of population growth and d is the rate of depreciation, then (d + n)k is break-eveninvestment,
23、 which is the amount necessary to keep constant the capital stockper worker k. Investment,break-eveninvestmentCapital per worker, k kk k*Break-eveninvestment, (d d + + n)k kInvestment, s s f(k k)For the economy to be in a steady state, investment s f(k) must offset the effects of depreciation and po
24、pulation growth (d + n)k. This is shown by the intersection of the two curves. An increase in the saving rate causes the capital stock to grow to a new steady state.Chapter Seven18Investment,break-eveninvestmentCapital per worker, k kk k*1Investment, s s f(k k)(d + d + n1)k kAn increase in the rate
25、of population growth shifts the line representing population growth and depreciation upward. The newsteady state has a lower level of capital per worker than theinitial steady state. Thus, the Solow model predicts that economies with higher rates of population growth will have lower levels of capita
26、l per worker and therefore lower incomes. k k* *2(d + d + n2)k kAn increase in the rate of population growth from n1 to n2 reduces the steady-state capital stock from k*1 to k*2.Chapter Seven19The change in the capital stock per worker is: D Dk k = i (d+d+n)k kNow, lets substitute sf(k k) for i: D D
27、k k = (sfk k) (d+d+n)k kThis equation shows how new investment, depreciation, and population growth influence the per-worker capital stock. New investment increases k, whereas depreciation and population growth decrease k. When we did not include the “n” variable in our simple versionwe were assumin
28、g a special case in which the population growth was 0.Chapter Seven20In the steady state, the positive effect of investment on the capital per worker just balances the negative effects of depreciation and population growth. Once the economy is in the steady state, investment has two purposes:1) Some
29、 of it, (dk*), replaces the depreciated capital, 2) The rest, (nk*), provides new workers with the steady state amount of capital.Capital per worker, k kk k*k k*The Steady StateInvestment, s f(k k)Break-even Investment,(d + (d + n) k kBreak-even investment, (d + (d + n) k kAn increase in the rate of
30、 growth of population will lower the level of output per worker.sf(k k)Chapter Seven21 In the long run, an economys saving determines the size of k and thus y. The higher the rate of saving, the higher the stock of capital and the higher the level of y. An increase in the rate of saving causes a per
31、iod of rapid growth,but eventually that growth slows as the new steady state is reached.Conclusion: although a high saving rate yields a highsteady-state level of output, saving by itself cannot generate persistent economic growth.Chapter Seven22Solow growth model Steady state Golden Rule level of capital
