Unformatted text preview: Notes Solow Growth Model Point of the class Math ! ! ! Logs and exponents Algebraic manipulation Partial derivatives and maximization Model building ! The Solow Growth Model Economic concepts and intuition ! ! ! ! Aggregate production function, CRS, diminishing returns, Cobb‐Douglas function Marginal product of capital and real interest rate Capital comes from saving output and is used to produce more output Diminishing returns causes capital accumulation to stop (at the steady state), so raising the saving rate only leads to temporary increases in the level of output per capita (during the transition period). Consumption per worker can be maximized by picking a saving rate at which the MPK=the depreciation rate (which equates the marginal benefit and the marginal cost of holding capital rather than consumption goods). The Solow Growth Model does not explain long‐run growth, however, because it leaves out technological progress ! ! Definitions Output per capita Aggregate Production Function State of technology (Total Factor Productivity) Constant returns to scale Capital‐per‐worker Real interest rate Capital accumulation function Decreasing returns to capital Cobb‐Douglas production function Saving rate Steady‐state Output‐per‐worker Marginal product of capital Depreciation rate Transition dynamics (or catch‐up to a common steady state) Convergence Golden‐rule level of capital Technological progress Homework Problems ! On the webpage. 1 Facts about growth See Notes for Economic Growth, Principles of Economics. Korea was able to achieve a much faster long‐run rate of growth than Nicragua. Why does GDP per worker increase? It would seem that it has a lot to do with the amount of capital that each worker in the economy gets to use, as we can see in the following graph: Pretty obviously, having more capital‐per‐worker helps. But if this were the whole explanation, the United States would be poorer than Finland or Switzerland, which is not the case. The extra output that can be obtained from extra capital is limited by diminishing returns. It turns out that improvements in technology are crucial. A measure of technology is Total Factor Productivity. What determines TFP? Things like human capital, research and development, and institutions. A combination of (physical and human) capital, denoted by K, and of the state of technology (denoted by A) is what drives living standards ! = "# $/% 0 Real GDP per capita 5000 10000 15000 20000 Facts 1950 1960 1970 1980 year 1990 2000 Real GDP per capita, Korea Real GDP per capita, Nicaragua Real GDP per worker (US=1), actual and predicted 0 .5 1 1.5 U.S.A. CANADA LUXEMBOURG AUSTRALIA NORWAY NETHERLANDS BELGIUM GERMANY, WEST ITALY SWEDEN NEW FRANCE ZEALAND AUSTRIA FINLAND ICELAND U.K. ISRAEL DENMARK SPAIN IRELAND VENEZUELA JAPAN SYRIA HONGMEXICO GREECE KONG ARGENTINA IRAN TAIWAN YUGOSLAVIA PORTUGALREP. KOREA, PANAMA CHILE ECUADOR COLOMBIA POLAND MAURITIUS PERU GUATEMALA DOMINICAN REP. TURKEY BOTSWANA MOROCCO LANKA PARAGUAY BOLIVIA SRI SWAZILAND THAILAND HONDURAS PHILIPPINES IVORYJAMAICA COAST ZIMBABWE NIGERIA INDIA SIERRA LEONE ZAMBIA NEPAL KENYA MADAGASCAR MALAWI SWITZERLAND 0 .5 1 1.5 Capital per worker (US=1) RGDP, predicted 2 Real GDP per worker, actual
Penn World Table 5.6. 1985 data U.S.A. .2 Total Factor Productivity .4 .6 .8 1 CANADA ICELANDNETHERLANDS AUSTRALIA ITALY U.K. LUXEMBOURG FRANCE BELGIUM SWEDEN NEW ZEALAND GERMANY, WEST NORWAY ISRAEL DENMARK AUSTRIA SPAIN SWITZERLAND HONG KONG MEXICO IRELAND FINLAND SYRIA VENEZUELA PARAGUAY ARGENTINA IRAN JAPAN YUGOSLAVIA GREECE MAURITIUS PORTUGAL CHILE GUATEMALA TAIWAN MOROCCO KOREA, REP. BOTSWANA DOMINICAN REP. SIERRA LEONE COLOMBIA PANAMA ECUADOR PERU TURKEY POLAND IVORY COAST SWAZILAND JAMAICA THAILAND BOLIVIA HONDURAS NIGERIA LANKA SRI PHILIPPINES NEPAL INDIA ZAMBIA KENYA ZIMBABWE MALAWI MADAGASCAR 0 .5 Real GDP per worker, 1985 1 Where does capital come from? The following is the Solow Growth Model, which endogenizes capital accumulation: it explains why countries accumulate capital, and the level of capital‐per‐worker at which they will reach long‐term equilibrium, or the steady state. See Robert Solow’s Nobel Autobiography at http://nobelprize.org/nobel_prizes/economics/laureates/1987/solow‐autobio.html 2 Model Building Simplifications ! ! ! ! ! ! ! A worker is a worker. Assume equal qualifications (homogenous labor) A unit of capital is a unit of capital, no matter what it actually is (homogenous capital) Technology is just the function that connects K and N to Y. An engineering blueprint. There’s no unemployment (or it doesn’t change) Constant returns to scale The economy is closed and the financial market works perfectly, so & = ' + (( ) * ). National saving equals investment. Saving is simply a proportion of income ' = +,. Building Blocks If we are trying to explain capital accumulation, we need to explain capital is accumulated (to produce output) and capital is accumulated (by saving and purchasing new capital goods). Aggregate production function Output is produced with capital and labor: that’s what capital is used for. Notice the role of A. This is the state of technology, which allows all the factors to be productive and determines output will be produced from given quantities of capital and labor. , = " (., /) We can pick almost any function to describe the relation between inputs and output … almost any function. Of course, the function must be positive (more inputs mean more output), and it would be nice if it were easy to work with. “Easy to work with” means that it should be monotonic (the relation shouldn’t ever change signs), continuous (no breaks), differentiable (no kinks). Diminishing returns to individual inputs A very important feature of the production function is that it must exhibit diminishing marginal returns to individual inputs 2, > ,0 =  (2., /) 2, > ,0 =  (2., /) So, for example, the production function , = "1. 12 Properties of exponents 3 4 ! 4 = (3!)4 3 4 3 5 = 3 4 65 34 = 3 4 75 35 (3 4 )5 = 3 45 3 exhibits diminishing marginal returns to individual inputs "12. 12 = "8121. 129 < 2"1. 12 The assumption of diminishing marginal returns to individual inputs is important for two reasons 1. It is an important characteristic of nearly any production process we can think of, once the process has been going on for a while, and 2. It is the key assumption of the Solow Model. Constant returns to scale CRS sometimes applies, sometimes it doesn’t. CRS means that increase factors will double production. While this may or may not be the case in reality, it makes the math much easier. And it might pay off to see whether we can sustain long‐run growth if returns to scale are constant (as opposed to, say, increasing returns to scale). 2, =  (2., 2/) The production function , = "1. 12 exhibits constant returns to scale "12. 122 = "812121. 129 = 2"1. 12 Output‐per‐worker and capital‐per‐worker CRS implies that if we divide the right‐hand side by a number, “N”, the left‐hand side will change by the same factor , ./ . = " : , ; = " : , 1; / // / From now on, = ! and = # . ! = "?(#, 1) = "?(# ) Suppose we double the number of workers and the number of units of capital. What happens to output? It doubles, by CRS. What happens to output per capita? Nothing. capital per worker, k C = 1D < @< @> @= 5000 =sqrt(5000) = = " A , B @= @= @= < = > = = We want the “F” function to exhibit both CRS and diminishing returns to K. We can ensure CRS by writing it as a “per worker function”, but we must pick a particular function to make it diminishing Diminishing returns to capital k "k "y 5000 70.71068 C = 1D 6000 1000 77.45967 6.748989 7000 1000 83.666 6.206336 8000 1000 89.44272 5.776716 4 returns. Log (! = "EFG(# )) and square root (! = "1#) both fit the bill. A special kind of function is the Cobb‐Douglas function, which turns out to be a very useful functional form. Cobb‐Douglas Production function , = ". H /$7H , 0 < I < 1 Diminishing returns to individual inputs 2, > ,0 = (2. , /) ,0 = "(2. )H /$7H = "2H . H /$7H , J = 2 H , < 2, Constant returns to scale 2, =  (2., 2/) , J = "(2. )H (2/)$7H = "2H (.)H 2$7H /$7H , J = "2H 2$7H (. )H /$7H = "2H6$7H (.)H /$7H = 2"(.)H /$7H , J = 2, Output per worker , ". H /$7H ". H /$7H = = H $7H / // / , . H / $7H = ": ; : ; / / / ! = "# H 5 capital per worker, k 0 1000 y=k0.45 0 22.38721 y=k0.5 0 31.62278 y=k0.55 0 44.66836 6 Saving Behavior Nations finance capital accumulation by giving up consumption. Output is either consumption goods or investment goods Income is either saved or consumed Combining these two equations, we get that 1 & // = '// How much do people save? Assume that saving is simply a proportion of income '// = +! Although saving rates (s) do vary over time and in response to the real interest rate, they don’t seem to be related to whether the country is poor or rich, and they don’t seem to change as a country grows richer. So we’ll take it as an exogenous parameter. Combining both equations, we get LM /N = OCM Investment Allocation equation K // + & // = ! K // + '// = ! Depreciation and Capital Accumulation Investment adds to capital. But some capital depreciates (wears out). Assume that a proportion # of capital wears out every period, so the capital that wears out is !" . Then next period’s capital is #P 6$ Q
4RSP TRUVWX Y Z > = #P [ \]ZP TRUVWX Y Z > + &P // Q 4R^ > T_U`a]ZRZ ) b #P c > Pa]P XVZ]TTR]UZ #P6$ ) #P = &P // ) b#P Capital Accumulation equation Time 0 1 2 3 4 5 kt 1000 1100 1190 1271 1343.9 1409.51 It/N 200 200 200 200 200 200 #kt 100 110 119 127.1 134.39 140.951 "kt 100 90 81 72.9 65.61 59.049 1 National Saving comes from households or firms (' TUVd]PR ) or from the government (( ) * ). So national saving ' = ' TUVd]PR + (( ) * ). If the financial market works perfectly, so that every unit saved gets lent out to productive uses, and total national saving equals total investment. An additional source of funds is foreigners, who send us their capital inflows (.& ), for example in the shape of foreign aid, loans, stock‐market purchases, or bank deposits. We’ll ignore foreigners for the moment. 7 Saving Behavior S/N = sy y 100 200 I/N =(0.5)y 50 100 I/N =(0.6)y 60 120 I/N=(0.7)y 70 140 Depreciation !"#
k 1000 2000 #k=(0.1)k$ 100 200 #k=(0.125)k$ 125 250 #k=(0.15)k$ 150 300 8 By how much does the capital stock change every period? It grows thanks to investment, which comes from people’s saving out of income; it shrinks due to depreciation. So we can combine the equation with the equation #P6$ = +!P + (1 ) b )#P to find the of capital‐per‐worker e#P = #P6$ ) #P = +!P ) b #P The Steady State We have an equation for how the capital stock changes. When does capital no longer change? 0 = e#P = +!P ) b #P +!P = b #P OCM f ghijk lmnomn7ojp7qlprjp = sDM c kjopjtuhnjk olpnulv lw lxk thounhx7ojp7qlprjp So capital accumulation stops when investment‐per‐worker (new capital‐per‐worker; the portion of output‐per‐worker that is saved) is equal to the portion of old capital‐per‐worker that depreciates away. This point is called the Steady State +!P = b #P +!P > b #P +!P < b #P capital stock is constant, capital stock grows, capital stock declines, e#P = 0, e#P > 0, e#P < 0, the economy is at the steady state the economy is below the steady state the economy is above the steady state Production Function y = Ak$/% Production function Labor and Technology For the moment, assume that both labor and technology are constant. 9 Unknowns/endogenous variables Equations 5. Allocation of Resources 3. Resource Constraint 4. Saving Function 2. Capital Accumulation 1. Production Function Parameters K // + '// = ! K // + &// = ! '// = +! #P6$ ) #P = &P // ) b#P H CM = "#P CM , DM6y , z/N, {/N, L/N  ~~~ " , / , + , b , / , #} Combining (1) and (3) implies K // = (1 ) +)!: consumption is whatever is left over, after households have decided to save. Combining (1) and (2) implies that saving is equal to investment. Combining (1), (2), and (3) tells us how investment is financed (it’s the investment allocation function) Putting the Building Blocks Together Solow Diagram The graph of !P with # in the horizontal axis (that is, the graph of the production function) would have to be a) upward sloping b) not a straight line, but a line that becomes flatter and flatter as we add #: extra units of capital produce less and less extra output. Suppose that the production function is !P = "#P . Then assume that " = 1, so that the equation that you graph is !P = #P . The graph of +!P , with # in the horizontal axis would have to be a) also upward sloping and exhibiting diminishing returns b) below the graph of the production function (if + < 100%, +! is less than !)/ If the production function is, then the investment function is & // = +!P = +#P . If we assume that the saving rate is + = 0.75, the equation that you graph is &// = 0.75#P . The graph of b #P , on the other hand, would have to be a straight line, since the depreciation rate doesn’t depend on the level of capital. Let’s assume b = 5%. Put them together, and you get the Solow Diagram 10 $/% $/% $/% $/% The production function and the investment function The depreciation function The Solow Diagram 11 The Principle of Transition Dynamics – “Catch up” e#P = +!P ) b #P % e#P = +"#P ) b#P $ e#P = 0.75"#P k y •Dy/€ 2.00 2.71 3 $/% ) 0.05#P #k •. •ƒD 0.4 1.0 1.35 2.90473751 y O•D€ M I •. ‚ƒ•Dy/€ 1.5 2.04 2.25 2.90473751 "k 8 20 27 ) sDM 1.1 1.04 0.9 0 58.09475019 3.872983346 ! 64 4 3 3.2 ‐0.2 125 5 3.75 6.25 ‐2.5 216 6 4.5 10.8 ‐6.3 When the capital stock is low, o it is highly productive (its MPK is very high) … which means a relatively high real interest rate, which attracts saving … which finances lots of investment on new capital, … which more than compensates the depreciation of capital o so capital accumulates ! When the capital stock is high, o it is not very productive (its MPK is rather low) … which means a relatively low real interest rate, which fails to attract saving … which leads to little investment on new capital, … which fails to maintain the level of capital as it depreciates, o so capital de‐accumulates ! When the capital stock is at the steady state, o the MPK is such that … the resulting real interest rate attracts just the right amount of saving … that generates just the right amount of investment … so that the amount of new capital is exactly equal to the amount of capital that depreciates o so the capital stock doesn’t change. 12 13 Steady State and its determinants „ How do we find the steady state? The definition of steady‐state capital‐per‐worker (denoted as #P ) is ek „ = 0 … Steady State Capital „ For this reason, steady‐state output‐per‐worker doesn’t change (!P ). „ ey… = 0 Steady State Output „ If the capital stock is not changing (e#P = 0), then „ „ „ e#P = +!P ) b #P = 0 General version „ „ „ e#P = +"?(#P ) ) b #P = 0 „ „ +"?(#P ) = b #P Cobb‐Douglas, I, version „ „ „ e#P = +"(#P )H ) b #P = 0 „ „ +"(#P )H = b #P +"† = #P‡ „ b ?(#P ) „ +"† = #P‡ „ = (# „ )$7H P b (#P )H
ˆ‰Š „ „ A+"†b B = ((#P )$7H )ˆ‰Š = (#P )ˆ‰Š ˆ ˆ ˆ ˆ‰Š „ „ „ +"?(#P ) = b #P „ #P ˆ‰Š = A+"†b B Steady‐state level of capital‐per‐worker is a) a positive function of the saving rate (+) b) a positive function of total factor productivity (") c) a positive function of the contribution of capital to production (I ). d) a negative function of the depreciation rate (b ). „ To find the steady‐state level of output‐per‐worker, we just plug #P into the production function: „ „ !P = "(#P )H „ !P
ˆH ZŒ ˆ‰Š " ‹A B Ž • = = ZŒ ˆ‰Š "A B • Š = "(")
Š Š ˆ‰Š Š Z ˆ‰Š AB • „ !P = „ "?(#P ) „ !P = (") ˆ ˆ‰Š ˆ‰Š A+†b B 14 Notice that this means that the steady‐state level of output‐per‐worker depends on the level of productivity. Not only is part of the production function, making capital more productive – it’s also a key determinant of capital‐per‐worker itself. And yet, as important as Total Factor Productivity is in the Solow Growth Model, it is left as an exogenous variable. The Capital/Output Ratio A useful concept is the “capital/output ratio”. It tells us how many units of capital are used to produce a unit of output. One of the reasons why it is such a useful concept is that it is (comparatively) easy to measure and to use in empirical tests of the theory. „ #P !P„ $7H A+"†b B $ (")$7H A+† $ „ #P • „ H !P $7H $7H A+†b B $7H A+†b B H $ „ „ #P #P + $7H = A+†b B • „ = !P„ !P b $7H = = • bB Interestingly, doesn’t depend on productivity (") or on %. That’s convenient, because it’s difficult to get empirical estimates of ". Testing the Solow Growth Model To test a model, we need to get it to generate a prediction. A simple, single, sharp prediction that involves actual data that we can get our hands on. It turns out that we can get our hands on the “capital/output ratio”, the ratio of the capital stock in an economy to GDP. We can also get our hands on the saving rate of an economy, and we can estimate the depreciation rate. The biggest shortcoming of the Solow model is that it doesn’t take into account total factor productivity ("). But since the Solow Model’s capital/output ratio doesn’t depend on A, the model should be able to predict it independently of the state of technology of the economy. That is, if the model cannot explain the data even after we’ve kept A out of the picture, it’s a pretty useless model indeed. But we will accept the Solow Growth Model if the data supports the idea that, more or less, „ #P + = !P„ b The problem of this little formula, though, is that it has variables on the right‐hand side. We can deal with this through multi‐variable regression, of course. On second thought, what if the depreciation rate is very similar across the world? Then there’s only one variable on the right‐hand side. Then the capital/output ratio should simply depend positively on the saving rate. This turns out to be true! 15 Convergence Suppose we have two economies with the same level of productivity ("), the same depreciation rate (b ), the same saving rate (+) and the same production function (governed by I). Then they must have the same steady‐state level capital‐per‐worker. „ #P $7H = A+"†b B $ If one of the economies has a lower level capital‐per‐worker, it is farther from its steady state, and it must be growing faster, on average. So if we plot the growth rate of countries that are pretty similar, such as the countries in the Organization for Economic Cooperation and Development, against the actual output‐per‐ capita a few decades ago, we should find that the countries that were poorest have had the highest average growth 16 rates. This is the principle of transition dynamics at work: it implies the convergence, or catch‐up over time, of the GDP‐ per‐capita of countries that have similar enough technologies. What about countries that have the same technology or production function? We would expect them to have different steady states. It’s perfectly plausible that, the economy of the United States and the economy of Zimbabwe are already at their steady states, so we would expect their average growth rates to be pretty darn close to independent of how rich they are. Comparative Statics in the Solow Growth Model 17 The Baseline I = 1/3, + = 0.75, b = 0.05, and " = 1 + " $7H 0.75(1) $7$ „ „ #P = : ; • #P = • ‘ % = (15)%/@ = 58.09 0.05 b
„ „ „ !P = "(#P )H • !P = "[58.09]H = 1[58.09]$/% = 3.87 $ $ Low Saving + = 0.375 The production function is as high as it was before: the MPK will behave just as it used to, capital will be just as productive. But now people aren’t saving as much per unit of output – they are not thinking about the future that much. They are perfectly content to stop capital accumulation sooner, which makes them poorer in the long run (but since before they were barely eating and now they get to eat more, perhaps they’ll be better off – see below). 0.375(1) $7$ „ #P = • ‘ % = (7.5)%/@ = 20.54 0.05
„ !P = (1)[20.54]% = 2.74 $ $ Low TFP " = 0.5, + = 0.75 The production function shifts down, so each level of capital‐per‐worker produces much less output‐per‐ worker, so there’s less available for saving and accumulating capital. Hence capital accumulation stops much sooner than in the baseline. A lower TFP makes workers less productive, so the marginal product of capital is smaller and investment in physical capital is less attractive. So less saving is attracted, and less capital is accumulated – which means that the point where new investment just barely manages to compensate for depreciation is reached earlier. 0.75(0.5) $7$ „ #P = • ‘ % = (7.5)%/@ = 20.54 0.05
„ !P = 0.5[20.54]$/% = 1.37 $ 18 Lower Saving Rate Lower State of Technology Baseline Low Saving Low TFP A 1.00 1.00 0.50 s 0.75 0.375 0.75 #$ 0.05 0.05 0.05 %$ 0.33 0.33 0.33 k* 58.09 20.54 20.54 y* 3.87 2.74 1.37 19 Is the effect of lowering the saving rate the same as the effect of lowering A? Steady‐state capital‐per‐ worker, though lower than in the baseline is the same as it was in the previous example. In the function, halving or halving give the same result: + " $7H „ #P = : ; b But changing TFP changes the production function and steady‐state capital‐per‐worker. Relatively unproductive workers have less capital‐per‐worker, so when TPF falls by 50%, steady‐state output falls by even more. „ „ !P = "(#P )H $ as in the low‐saving case: the production function is lower than previously, so this same level of is less productive. Output per worker is lower, even though the level of capital is the same. Low s b = 0.04, + = 0.75, " = 1 Because the capital stock depreciates more slowly, capital‐per‐worker keeps growing for a longer time before it stops. „ #P = (18.75)%/@ = 81.19 „ !P = (1)[81.19]% = 4.33 $ Low ’ I = 0.25, b = 0.05, + = 0.75, " = 1 Remember that the formula for the Marginal Productivity of Capital we found above. In the Cobb‐ Douglas function, , “”. = I . . When % contracts, capital becomes less productive: the production function shifts in and it becomes harder to attract savings. People are less likely to pass up current consumption and buy (now less productive) capital, so capital accumulation stops sooner. „ #P = $ $ $7 (2.5) • = (15)% = 36.99 $ • „ „ !P = "(#P )H = (1)[36.99]• = 2.47 20 Lower Depreciation Rate Lower Contribution of Capital Baseline Low delta Low alpha A 1.00 1.00 1.00 s 0.75 0.75 0.75 #$ 0.05 0.04 0.05 %$ 0.33 0.33 0.25 k* 58.09 81.19 36.99 y* 3.87 4.33 2.47 21 Steady State and Saving 1. The saving rate is positively correlated with steady‐state capital‐per‐worker and steady‐state output‐per‐worker. The formula that defines the steady‐state level of capital‐per‐worker and steady‐state level of output ‐per‐worker shows that these two values depend on the saving rate. Higher levels of saving generate higher steady states; less saving means lower steady states. Experiment with different levels of the saving rate Notice that the intersection of the saving curve with the depreciation curve happens sooner if the saving rate is smaller: steady‐state capital‐per‐worker is lower if the saving rate is lower. This means that steady‐state output‐per‐worker is lower. s k* y* 0.80 64.00 4.00 0.75 58.09 3.87 0.50 31.62 3.16 0.25 11.18 2.24 0.10 2.83 1.41 A high saving rate suggests that economic agents value future consumption relatively more than they value current consumption. This “thrifty” behavior allows them to have very tight belts today but a large amount of consumption in the future. 22 2. The saving rate has no effect on the growth rate of output in the steady state. In the steady state, capital‐per‐worker doesn’t change. That follows from the definition of the steady state. „ e#P = 0 For the same reason, output‐per‐worker doesn’t change. „ e!P = 0 So what happens to the of output in the steady state if the saving rate changes? Nothing. Output becomes steady, so its growth rate is zero. Between steady states, however… y1 Output per worker , y With saving rate s1>s0 y0 With saving rate s0 Time Although the level of output in the steady state depends on the saving rate, its growth rate doesn’t. The reason is diminishing returns to capital. Dedicating a greater proportion of income to capital accumulation doesn’t change the fact that, eventually, capital stops being very productive. 3. But changing the saving rate affects the growth rate of output in the transition to the steady state. Increasing the saving rate gives an economy more new capital per unit of old capital, so it manages to keep depreciation at bay for a longer time. Imagine two countries that have identical technologies and that start out with the same level of capital‐per‐worker. Both country A and country B take their existing (for the moment identical) amount of capital, produce (identical) output with it. The countries then save some output – 23 but because at the same time, some of the capital depreciates, not all the saving goes to new capital. Some of it merely replaces the worn out capital. The only difference between country A and country B is their saving rates. Country A saves a greater proportion of its income, perhaps because it has a better financial system, one that makes it easier and safer to save rather than to spend thoughtlessly. Period 0 1 2 kt 10.000 11.116 12.234 y 2.154 2.232 2.304 0.75 yt 1.616 1.674 1.728 #kt$ 0.500 0.556 0.612 "kt$ 1.116 1.118 1.116 99 100 0 1 2 57.000 58.090 59.000 kt 10.000 10.577 11.146 3.849 3.873 3.893 y 2.154 2.195 2.234 2.886 2.905 2.920 0.50 yt 1.077 1.098 1.117 2.850 2.905 2.950 #kt$ 0.500 0.529 0.557 0.036 0.000 ‐0.030 "kt$ 0.577 0.569 0.560 99 31.000 3.141 1.571 1.550 0.021 100 Country A will have more 31.620 3.162 1.581 1.581 0.000 capital left over after 32.000 3.175 1.587 1.600 ‐0.013 depreciation to put in more new capital, which allows it to continue growing. Diminishing returns eventually will stop growth, but at a higher . a function its derivative 24 Allocating Resources How do firms decide how much capital and how much labor to use? They maximize their profits. Their profits might be given by a profit function such as this, ” (., 2) ) –. ) —/ which simply says that profit is the difference between revenue (price times output) minus costs (rental for capital, wage for labor) So the firm maximizes this function by choosing capital and labor. General version max>,= [” (., 2) ) –. ) —/] A little bit of Calculus: the Power Rule ?(3 ) = Ÿ3 4 ?(3 ) = ?0(3 ) = Ÿ¡3 47$ 3 ?(3 ) = 23 @ + 33 + 5 ?(3 ) = 23 @ + 33 $ + 53 } ? J (3) = 2(23 @7$ ) + 3(13 $7$ ) + 5(03 }7$ ) ?0(3 ) = 43 $ + 3 Cobb‐Douglas, I = 1/3, version max>,= [”". H /$7H ) –. ) —/] max>,= ˜”".$/% / @/% ) –. ) —/™ To do that, we simply find the slope of the profit function (the derivative) and find where the slope =0. That would be where the function reaches a maximum. General version Profit =  (. , /) ) –. ) —/ š› (>,œ ) )– š> Cobb‐Douglas, I = 1/3, version Profit = ". $/% / @/% ) –. ) —2 $ ". $/%7$ / @/% % = 0 ) – = 0 = – “”. = – $ >ˆ/• @/% ">/ % $ Œ>ˆ/• = ž/• > % = – = – k 1 8 27 64 125 216 MPK 0.333 0.083 0.037 0.021 0.013 0.009 25 “”. = $< %> So for a Cobb‐Douglas function, the marginal product of capital is proportional to the average amount of output produced by . (that is, ,/.), and the factor of proportionality is I = 1/3. In plainer English: the productivity of capital depends on whether there’s a lot of it or a little of it. How about the marginal productivity of labor? Take a derivative of the profit function with respect to labor and set it equal to zero š› (>,=) ) š= — = 0 @ ". $/% / @/%7$ % @< %= ) — = 0 “”2 = — = — So the marginal product of labor is proportional to the average amount of output produced by /, where the factor of proportionality is (1 ) I ) = 2/3. We can use the formula for the MPK that we derived above to calculate the capital share of output, which would be – (the income earned by capital) times . (the capital stock) divided by , (in order to express it as a %). Then, the capital share is equal to … I. “”. = I , . = – • I = – . ,
= < So the labor share of output is (1 ) I ), because (1 ) I ) = — . In practice, the share of GDP earned by owners of capital is about 1/3, and the share of wages in GDP is about 2/3 … for most countries in most time periods. So we’ll keep using I = 1/3. If the value of a company in the financial market is the value that savers give to their expected income from the stock of capital, then the fundamental value of the stock market should be related to –, ./,, and I . The Real Interest Rate The and saving one unit of output. A unit of saving is used as a unit of investment, which is a new unit of capital, which produces an extra MPK of output. So the income that can be earned from saving – the real interest rate – is equal to the marginal product of capital. “”. = – “”. > – capital stock at its optimal firms can borrow more and earn MPK above the borrowing cost. As they borrow more, they drive the real interest rate up. firms that borrow to buy capital find that the borrowing cost exceeds the returns from buying capital. As they borrow less, they drive the real interest rate down. “”. < – 26 The Optimal Level of Saving – the Golden Rule More saving means more capital accumulation and more output. What about consumption? A country that saves will have a lot of output, but won’t eat a whole lot – consumption will be very low. A country that saves a little will have little output, will eat almost all of it … and consumption will be very low, too. Viewed in a different way, a society could choose to increase its consumption for today – have a big nice party, at the expense of consumption future generations. Or a society could also choose allow for more consumption for future generations, but only by reducing today’s consumption. Remember that consumption‐per‐worker is ' K =!) / / This is also true at the steady state. '„ K„ = !„ ) / / The level of consumption‐per‐ worker is different at different steady states, which are determined by the different saving rates. We want to know the saving rate that gives the optimal level of consumption. Choosing an “optimum” means choosing Steady‐State Capital‐ per‐worker k* 89.44 76.37 64.00 52.38 41.57 31.62 22.63 14.70 8.00 2.83 0.00 Steady‐State Output‐per‐ worker y* 4.47 4.24 4.00 3.74 3.46 3.16 2.83 2.45 2.00 1.41 0.00 Steady‐State Saving‐ per‐worker S*/N 4.47 3.82 3.20 2.62 2.08 1.58 1.13 0.73 0.40 0.14 0.00 Steady‐State Consumption‐ per‐worker C*/N 0.00 0.42 0.80 1.12 1.39 1.58 1.70 1.71 1.60 1.27 0.00 Saving Rate s 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 optimum saving rate ! optimum steady‐state capital‐per‐worker ! optimum steady‐state output‐per‐worker ! biggest steady‐state consumption‐per‐worker At the “optimum”, any change must make society worse off. The optimum point would be such that , present and future, is better off than in any other point. is that which gives the same level of „ „ consumption to current and future generations. Optimal consumption (and therefore optimal !P and #P and ) is that which makes everyone best‐off. 27 The graph below shows three different economies in their steady states. (Notice the three economies depicted are in the steady state). If consumption is the difference between output and saving, and if output is denoted by the blue line while saving (and investment) are denoted by the red line (or the maroon or the orange lines), the vertical distance between the two lines must be equal consumption (at any level of capital). Low levels of saving produce little output – which is consumed almost entirely. High levels of saving produce a lot of output, but little of it is consumed. Somewhere in the middle there’s a saving rate that gives maximum consumption. If steady‐state consumption‐per‐worker is income‐per‐worker minus saving‐per‐worker and saving per worker is a proportion of income, while income is determined by the production function 28 K„ K„ „ „ „ „ = !P ) +!P • = ?(#P ) ) +?(#P ) / /
„ „ „ In the steady state, e#P = 0 means that +?(#P ) = b #P . Then steady‐state consumption‐per‐worker is given by K„ „ „ = ?(#P ) ) b#P / We know that the saving rate is positively related to the steady‐state capital‐per‐worker. So we can „ „ focus on finding DM . We do this by taking a derivative of the above function with respect to #P and setting it equal to zero. K„ / = ?0(# „ ) ) b = 0 P „ #P
„ ?0(#P ) = b “”. = b This tells us that the consumption‐maximizing steady state is one were the MPK is equal to the rate of depreciation. This is a basic “micro” conclusion: marginal benefit must equal marginal cost. What is the benefit of owning capital? You get to produce output. What is the benefit of an of capital? The extra output, the MPK. So the MPK is the benefit of giving up some current consumption to purchase a long‐lived asset for the future. What is the cost of owning capital? If you had partied away your wealth at least you’d have had the good times. But if you hold capital, after a while you get a rusty, moth‐eaten bit of junk. Capital decays. What is the cost of owning an extra bit of steady‐state capital‐per‐worker? Its wear‐and‐tear, its depreciation. So the cost of saving a bit more to accumulate one more unit of capital is the depreciation rate. So we forgo consumption until the marginal benefit of holding capital (the MPK) is equal to the marginal cost of holding capital (the depreciation rate). That is the optimal amount of rate of saving, of foregoing consumption. Notice also that MPK is the slope of the production function: the additional output‐per‐worker out of the additional capital‐per‐worker. b is, well, the slope of the depreciation function. So the largest consumption‐per‐worker (the largest distance between the amount produced and the amount saved in the steady state) is found at the level of capital where the production function and the depreciation function are parallel. “”. = b Slope of production function = slope of depreciation function 29 30 For example, for the Cobb‐Douglas function “”. „ = I
„ !P „ #P If the consumption‐maximizing steady‐stead level of capital‐per‐worker is given by “”. = b , then the golden‐rule level of steady‐state capital per worker # ¢£ and the golden‐rule output per worker ! ¢£ are given by I ! ¢£ = b # ¢£ I! ¢£ = b# ¢£ „ In the steady state, b #P = +! „. Is this true at the Golden Rule level? Of course! It’s the golden‐rule level of capital per worker. This is just steady state. Then we can say, b # ¢£ = + ¢£ ! ¢£ . I! ¢£ = + ¢£ ! ¢£ + ¢£ is what we are looking for, the consumption‐maximizing saving rate I = + ¢£ In the specific case of I = 1/3, + ¢£ = . So if the capital contribution to output (%) is about one‐third, more or less, the golden‐rule saving rate + ¢£ should average one‐third. Numerical exercise Suppose that " = 1, and b = 5% and I = 25%. Show that, at the Golden‐Rule saving rate, consumption‐per‐worker is maximized. We know that the Golden Rule says that I = + ¢£ and so in this case + ¢£ = 0.25. Given the parameters, let’s first find the MPK (make sure that you understand the derivation) +" $7H " ¤A B ¥ b +" AB b
$ $7H $ H $ % “”. „ = I „ !P „=I #P = I" +" $7H ¤A b B ¥
$ $7H = I" Ib = +" + AB b “”. „ = (0.25)0.05 1 = (0.0125) + + It will be handy to have formulas for steady‐state capital‐per‐worker, steady‐state output‐per‐worker, and steady‐state consumption‐per‐worker. 31 + " $7H +(1) $7}[email protected]¦ „ #P = : ; =• = (+)•§% (20)•§% ‘ (0.05) b
„ !P $ $ = ˆH ZŒ ˆ‰Š " ‹A B Ž • = (1)˜(+)•§% (20)•§% ™ }[email protected]¦ = ˜(+)•§% (20)•§% ™ $/• = (+)$§% (20)$§% z„ §N „ ) +!P 1.134 1.282 1.200 Saving Rate 0.10 0.25 0.40 MPK (1§+)(0.0125) 0.125 0.050 0.031 (+ D„ M •§% ( ) 20 2.520 8.550 16.000 )•§% (+ C„ M $§% ( ) 20 1.260 1.710 2.000 )$§% „ !P Increasing its saving rate makes this society’s steady‐state capital‐per‐worker grow dramatically. Fast‐ rising capital must be maintained, and that means that this country is devoting fast‐rising amounts of resources to fighting off depreciation. On the other hand, due to diminishing returns, output is not rising that fast. At pretty low levels of capital, there’s not much capital to maintain, so most output goes to consumption. This means that a growing share of (slow‐growing) output goes to pay for (fast‐rising) depreciation, leaving less for consumption. Notice how, in the low‐saving economy, the MPK is very high, much higher than the depreciation rate. Recall that this means that the marginal benefit of holding an additional unit of capital exceeds the marginal cost of that unit of capital – it would make sense for this economy to hold more capital, to save more. But the citizens of this economy are short‐sighted and prefer to consume today. Because they don’t save much, there are so few units of (high‐productivity) capital that the overall return is barely enough to offset depreciation and the economy is at the steady state. In the high‐saving economy, on the other hand, the MPK is very low, far below the depreciation rate. This society is holding so much capital that its marginal benefit doesn’t justify the marginal cost – it would make sense for this economy to save less. Nevertheless, the citizens forego so much consumption that they manage to offset the depreciation. For this reason, in both the low‐saving and the high‐saving economies consumption is lower than it could be if the society brought its saving rate to equality with I , that is, to 25%. 32 Strengths and Weaknesses of the Solow Growth Model Strengths It explains why countries are rich or poor in the very long run ! ! ! Total Factor Productivity (related perhaps to education, legal environment, etc.) Saving and investment (related perhaps to culture and the quality of the financial system) Low rate of depreciation (related perhaps to weather or the quality of machinery) It explains why growth rates differ between countries with similar steady states ! ! Countries that are closer to the steady‐state (like the US) grow more slowly Countries that are farther away from the steady‐state (like Ireland) grow more quickly Weaknesses Leaves saving rates as exogenous. Saving rates are probably related to how well the financial system functions, how patient people are, or how the tax system punishes or rewards saving. A full model would explain why people make individual choices that are (or aren’t) consistent with the optimum. Leaves Total Factor Productivity as exogenous. Because capital accumulation lead to long‐run growth (eventually, output growth stops in the steady state), the Solow Growth Model is not a theory of long‐run growth, but a theory of transition dynamics. 33 ...
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 Spring '10
 Williamson
 Macroeconomics, gross domestic product, MPK, saving rate

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