If you have ever spoken with your grandparents about what their lives were like


 Harold Caldwell
 2 years ago
 Views:
Transcription
1 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth The question of growth is nothing new but a new isguise for an ageol issue, one which has always intrigue an preoccupie economics: the present versus the future. James Tobin If you have ever spoken with your granparents about what their lives were like when they were young, most likely you learne an important lesson about economic growth: material stanars of living have improve substantially over time for most families in most countries. This avance comes from rising incomes, which have allowe people to consume greater quantities of goos an services. To measure economic growth, economists use ata on gross omestic prouct, which measures the total income of everyone in the economy. The real GDP of the Unite States toay is more than five times its 1950 level, an real GDP per person is more than three times its 1950 level. In any given year, we also observe large ifferences in the stanar of living among countries. Table 71 shows the 2007 income per person in the worl s 14 most populous countries. The Unite States tops the list with an income of $45,790 per person. Banglaesh has an income per person of only $1,242 less than 3 percent of the figure for the Unite States. Our goal in this part of the book is to unerstan what causes these ifferences in income over time an across countries. In Chapter 3 we ientifie the factors of prouction capital an labor an the prouction technology as the sources of the economy s output an, thus, of its total income. Differences in income, then, must come from ifferences in capital, labor, an technology. Our primary task in this chapter an the next is to evelop a theory of economic growth calle the Solow growth moel. Our analysis in Chapter 3 enable us to escribe how the economy prouces an uses its output at one point in time. The analysis was static a snapshot of the economy. To explain why our national income grows, an why some economies grow faster than others, we must broaen our analysis so that it escribes changes in the economy over time. By eveloping such a moel, we make our analysis ynamic more like a 191
2 192 PART III Growth Theory: The Economy in the Very Long Run TABLE 71 Income per Income per Country person (2007) Country person (2007) Unite States $45,790 Inonesia 3,728 Japan 33,525 Philippines 3,410 Germany 33,154 Inia 2,753 Russia 14,743 Vietnam 2,600 Mexico 12,780 Pakistan 2,525 Brazil 9,570 Nigeria 1,977 China 5,345 Banglaesh 1,242 Source: The Worl Bank. International Differences in the Stanar of Living movie than a photograph. The Solow growth moel shows how saving, population growth, an technological progress affect the level of an economy s output an its growth over time. In this chapter we analyze the roles of saving an population growth. In the next chapter we introuce technological progress The Accumulation of Capital The Solow growth moel is esigne to show how growth in the capital stock, growth in the labor force, an avances in technology interact in an economy as well as how they affect a nation s total output of goos an services. We will buil this moel in a series of steps. Our first step is to examine how the supply an eman for goos etermine the accumulation of capital. In this first step, we assume that the labor force an technology are fixe. We then relax these assumptions by introucing changes in the labor force later in this chapter an by introucing changes in technology in the next. The Supply an Deman for Goos The supply an eman for goos playe a central role in our static moel of the close economy in Chapter 3. The same is true for the Solow moel. By consiering the supply an eman for goos, we can see what etermines how 1 The Solow growth moel is name after economist Robert Solow an was evelope in the 1950s an 1960s. In 1987 Solow won the Nobel Prize in economics for his work on economic growth. The moel was introuce in Robert M. Solow, A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics (February 1956):
3 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 193 much output is prouce at any given time an how this output is allocate among alternative uses. The Supply of Goos an the Prouction Function The supply of goos in the Solow moel is base on the prouction function, which states that output epens on the capital stock an the labor force: Y = F(K, L). The Solow growth moel assumes that the prouction function has constant returns to scale. This assumption is often consiere realistic, an, as we will see shortly, it helps simplify the analysis. Recall that a prouction function has constant returns to scale if zy = F(zK, zl) for any positive number z. That is, if both capital an labor are multiplie by z, the amount of output is also multiplie by z. Prouction functions with constant returns to scale allow us to analyze all quantities in the economy relative to the size of the labor force. To see that this is true, set z = 1/L in the preceing equation to obtain Y/L = F(K/L, 1). This equation shows that the amount of output per worker Y/L is a function of the amount of capital per worker K/L. (The number 1 is constant an thus can be ignore.) The assumption of constant returns to scale implies that the size of the economy as measure by the number of workers oes not affect the relationship between output per worker an capital per worker. Because the size of the economy oes not matter, it will prove convenient to enote all quantities in per worker terms. We esignate quantities per worker with lowercase letters, so y = Y/L is output per worker, an k = K/L is capital per worker. We can then write the prouction function as y = f(k), where we efine f(k) = F(k, 1). Figure 71 illustrates this prouction function. The slope of this prouction function shows how much extra output a worker prouces when given an extra unit of capital. This amount is the marginal prouct of capital MPK. Mathematically, we write MPK = f(k + 1) f(k). Note that in Figure 71, as the amount of capital increases, the prouction function becomes flatter, inicating that the prouction function exhibits iminishing marginal prouct of capital. When k is low, the average worker has only a little capital to work with, so an extra unit of capital is very useful an prouces a lot of aitional output. When k is high, the average worker has a lot of capital alreay, so an extra unit increases prouction only slightly.
4 194 PART III Growth Theory: The Economy in the Very Long Run FIGURE 71 Output per worker, y 1 MPK Output, f(k) The Prouction Function The prouction function shows how the amount of capital per worker k etermines the amount of output per worker y = f(k). The slope of the prouction function is the marginal prouct of capital: if k increases by 1 unit, y increases by MPK units. The prouction function becomes flatter as k increases, inicating iminishing marginal prouct of capital. Capital per worker, k The Deman for Goos an the Consumption Function The eman for goos in the Solow moel comes from consumption an investment. In other wors, output per worker y is ivie between consumption per worker c an investment per worker i: y = c + i. This equation is the perworker version of the national income accounts ientity for an economy. Notice that it omits government purchases (which for present purposes we can ignore) an net exports (because we are assuming a close economy). The Solow moel assumes that each year people save a fraction s of their income an consume a fraction (1 s). We can express this iea with the following consumption function: c = (1 s)y, where s, the saving rate, is a number between zero an one. Keep in min that various government policies can potentially influence a nation s saving rate, so one of our goals is to fin what saving rate is esirable. For now, however, we just take the saving rate s as given. To see what this consumption function implies for investment, substitute (1 s)y for c in the national income accounts ientity: Rearrange the terms to obtain y = (1 s)y + i. i = sy.
5 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 195 This equation shows that investment equals saving, as we first saw in Chapter 3. Thus, the rate of saving s is also the fraction of output evote to investment. We have now introuce the two main ingreients of the Solow moel the prouction function an the consumption function which escribe the economy at any moment in time. For any given capital stock k, the prouction function y = f(k) etermines how much output the economy prouces, an the saving rate s etermines the allocation of that output between consumption an investment. Growth in the Capital Stock an the Steay State At any moment, the capital stock is a key eterminant of the economy s output, but the capital stock can change over time, an those changes can lea to economic growth. In particular, two forces influence the capital stock: investment an epreciation. Investment is expeniture on new plant an equipment, an it causes the capital stock to rise. Depreciation is the wearing out of ol capital, an it causes the capital stock to fall. Let s consier each of these forces in turn. As we have alreay note, investment per worker i equals sy. By substituting the prouction function for y, we can express investment per worker as a function of the capital stock per worker: i = sf(k). This equation relates the existing stock of capital k to the accumulation of new capital i. Figure 72 shows this relationship. This figure illustrates how, for any value of k, the amount of output is etermine by the prouction function f(k), FIGURE 72 Output per worker, y Output per worker Output, f(k) Output, Consumption, an Investment The saving rate s etermines the allocation of output between consumption an investment. For any level of capital k, output is f(k), investment is sf(k), an consumption is f(k) sf(k). y c Consumption per worker Investment, sf(k) i Investment per worker Capital per worker, k
6 196 PART III Growth Theory: The Economy in the Very Long Run an the allocation of that output between consumption an saving is etermine by the saving rate s. To incorporate epreciation into the moel, we assume that a certain fraction of the capital stock wears out each year. Here (the lowercase Greek letter elta) is calle the epreciation rate. For example, if capital lasts an average of 25 years, then the epreciation rate is 4 percent per year ( = 0.04). The amount of capital that epreciates each year is k. Figure 73 shows how the amount of epreciation epens on the capital stock. We can express the impact of investment an epreciation on the capital stock with this equation: Change in Capital Stock = Investment Depreciation D k = i k, where k is the change in the capital stock between one year an the next. Because Dinvestment i equals sf(k), we can write this as D k = sf(k) k. Figure 74 graphs the terms of this equation investment an epreciation for ifferent levels of the capital stock k. The higher the capital stock, the greater the amounts of output an investment. Yet the higher the capital stock, the greater also the amount of epreciation. As Figure 74 shows, there is a single capital stock k* at which the amount of investment equals the amount of epreciation. If the economy fins itself at this level of the capital stock, the capital stock will not change because the two forces acting on it investment an epreciation just balance. That is, at k*, D k = 0, so the capital stock k an output f(k) are steay over time (rather than growing or shrinking). We therefore call k* the steaystate level of capital. The steay state is significant for two reasons. As we have just seen, an economy at the steay state will stay there. In aition, an just as important, FIGURE 73 Depreciation per worker, k Depreciation, k Depreciation A constant fraction of the capital stock wears out every year. Depreciation is therefore proportional to the capital stock. Capital per worker, k
7 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 197 FIGURE 74 Investment an epreciation k 2 i 2 i* k* i 1 k 1 Depreciation, k Investment, sf(k) Investment, Depreciation, an the Steay State The steaystate level of capital k* is the level at which investment equals epreciation, inicating that the amount of capital will not change over time. Below k* investment excees epreciation, so the capital stock grows. Above k* investment is less than epreciation, so the capital stock shrinks. Capital stock increases because investment excees epreciation. k 1 k* k 2 Capital per worker, k Steaystate level of capital per worker Capital stock ecreases because epreciation excees investment. an economy not at the steay state will go there. That is, regarless of the level of capital with which the economy begins, it ens up with the steaystate level of capital. In this sense, the steay state represents the longrun equilibrium of the economy. To see why an economy always ens up at the steay state, suppose that the economy starts with less than the steaystate level of capital, such as level k 1 in Figure 74. In this case, the level of investment excees the amount of epreciation. Over time, the capital stock will rise an will continue to rise along with output f(k) until it approaches the steay state k*. Similarly, suppose that the economy starts with more than the steaystate level of capital, such as level k 2. In this case, investment is less than epreciation: capital is wearing out faster than it is being replace. The capital stock will fall, again approaching the steaystate level. Once the capital stock reaches the steay state, investment equals epreciation, an there is no pressure for the capital stock to either increase or ecrease. Approaching the Steay State: A Numerical Example Let s use a numerical example to see how the Solow moel works an how the economy approaches the steay state. For this example, we assume that the prouction function is Y = K 1/2 L 1/2. From Chapter 3, you will recognize this as the Cobb Douglas prouction function with the capitalshare parameter equal to 1/2. To erive the perworker a
8 198 PART III Growth Theory: The Economy in the Very Long Run prouction function f(k), ivie both sies of the prouction function by the labor force L: Y K = 1/2 L 1/2. L L Rearrange to obtain Y L K L = ( ) 1/2. Because y = Y/L an k = K/L, this equation becomes y = k 1/2, which can also be written as y = k. This form of the prouction function states that output per worker equals the square root of the amount of capital per worker. To complete the example, let s assume that 30 percent of output is save (s = 0.3), that 10 percent of the capital stock epreciates every year ( = 0.1), an that the economy starts off with 4 units of capital per worker (k = 4). Given these numbers, we can now examine what happens to this economy over time. We begin by looking at the prouction an allocation of output in the first year, when the economy has 4 units of capital per worker. Here are the steps we follow. Accoring to the prouction function y = k, the 4 units of capital per worker (k) prouce 2 units of output per worker (y). Because 30 percent of output is save an investe an 70 percent is consume, i = 0.6 an c = 1.4. Because 10 percent of the capital stock epreciates, k = 0.4. With investment of 0.6 an epreciation of 0.4, the change in the capital stock is D k = 0.2. Thus, the economy begins its secon year with 4.2 units of capital per worker. We can o the same calculations for each subsequent year. Table 72 shows how the economy progresses. Every year, because investment excees epreciation, new capital is ae an output grows. Over many years, the economy approaches a steay state with 9 units of capital per worker. In this steay state, investment of 0.9 exactly offsets epreciation of 0.9, so the capital stock an output are no longer growing. Following the progress of the economy for many years is one way to fin the steaystate capital stock, but there is another way that requires fewer calculations. Recall that D k = sf(k) k. This equation shows how k evolves over time. Because the steay state is (by efinition) the value of k at which k = 0, we know that D 0 = sf(k*) k*,
9 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 199 TABLE 72 Approaching the Steay State: A Numerical Example Assumptions: y = k ; s = 0.3; = 0.1; initial k = 4.0 Year k y c i k D k or, equivalently, k* f(k*) s =. This equation provies a way of fining the steaystate level of capital per worker, k*. Substituting in the numbers an prouction function from our example, we obtain k* 0.3 =. k * 0.1 Now square both sies of this equation to fin k* = 9. The steaystate capital stock is 9 units per worker. This result confirms the calculation of the steay state in Table 72.
10 200 PART III Growth Theory: The Economy in the Very Long Run CASE STUDY The Miracle of Japanese an German Growth Japan an Germany are two success stories of economic growth. Although toay they are economic superpowers, in 1945 the economies of both countries were in shambles. Worl War II ha estroye much of their capital stocks. In the ecaes after the war, however, these two countries experience some of the most rapi growth rates on recor. Between 1948 an 1972, output per person grew at 8.2 percent per year in Japan an 5.7 percent per year in Germany, compare to only 2.2 percent per year in the Unite States. Are the postwar experiences of Japan an Germany so surprising from the stanpoint of the Solow growth moel? Consier an economy in steay state. Now suppose that a war estroys some of the capital stock. (That is, suppose the capital stock rops from k* to k 1 in Figure 74.) Not surprisingly, the level of output falls immeiately. But if the saving rate the fraction of output evote to saving an investment is unchange, the economy will then experience a perio of high growth. Output grows because, at the lower capital stock, more capital is ae by investment than is remove by epreciation. This high growth continues until the economy approaches its former steay state. Hence, although estroying part of the capital stock immeiately reuces output, it is followe by higherthannormal growth. The miracle of rapi growth in Japan an Germany, as it is often escribe in the business press, is what the Solow moel preicts for countries in which war has greatly reuce the capital stock. How Saving Affects Growth The explanation of Japanese an German growth after Worl War II is not quite as simple as suggeste in the preceing case stuy. Another relevant fact is that both Japan an Germany save an invest a higher fraction of their output than oes the Unite States. To unerstan more fully the international ifferences in economic performance, we must consier the effects of ifferent saving rates. Consier what happens to an economy when its saving rate increases. Figure 75 shows such a change. The economy is assume to begin in a steay state with saving rate s 1 an capital stock k* 1. When the saving rate increases from s 1 to s 2, the sf(k) curve shifts upwar. At the initial saving rate s 1 an the initial capital stock k* 1, the amount of investment just offsets the amount of epreciation. Immeiately after the saving rate rises, investment is higher, but the capital stock an epreciation are unchange. Therefore, investment excees epreciation. The capital stock will graually rise until the economy reaches the new steay state k* 2, which has a higher capital stock an a higher level of output than the ol steay state. The Solow moel shows that the saving rate is a key eterminant of the steaystate capital stock. If the saving rate is high, the economy will have a large capital stock an a high level of output in the steay state. If the saving rate is low, the econ
11 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 201 FIGURE 75 Investment an epreciation k s 2 f(k) s 1 f(k) causing the capital stock to grow towar a new steay state. 1. An increase in the saving rate raises investment,... k* 1 An Increase in the Saving Rate An increase in the saving rate s implies that the amount of investment for any given capital stock is higher. It therefore shifts the saving function upwar. At the initial steay state k 1 *, investment now excees epreciation. The capital stock rises until the economy reaches a new steay state k 2 * with more capital an output. k* 2 Capital per worker, k omy will have a small capital stock an a low level of output in the steay state. This conclusion shes light on many iscussions of fiscal policy. As we saw in Chapter 3, a government buget eficit can reuce national saving an crow out investment. Now we can see that the longrun consequences of a reuce saving rate are a lower capital stock an lower national income. This is why many economists are critical of persistent buget eficits. What oes the Solow moel say about the relationship between saving an economic growth? Higher saving leas to faster growth in the Solow moel, but only temporarily. An increase in the rate of saving raises growth only until the economy reaches the new steay state. If the economy maintains a high saving rate, it will maintain a large capital stock an a high level of output, but it will not maintain a high rate of growth forever. Policies that alter the steaystate growth rate of income per person are sai to have a growth effect; we will see examples of such policies in the next chapter. By contrast, a higher saving rate is sai to have a level effect, because only the level of income per person not its growth rate is influence by the saving rate in the steay state. Now that we unerstan how saving an growth interact, we can more fully explain the impressive economic performance of Germany an Japan after Worl War II. Not only were their initial capital stocks low because of the war, but their steaystate capital stocks were also high because of their high saving rates. Both of these facts help explain the rapi growth of these two countries in the 1950s an 1960s.
12 202 PART III Growth Theory: The Economy in the Very Long Run CASE STUDY Saving an Investment Aroun the Worl We starte this chapter with an important question: Why are some countries so rich while others are mire in poverty? Our analysis has taken us a step closer to the answer. Accoring to the Solow moel, if a nation evotes a large fraction of its income to saving an investment, it will have a high steaystate capital stock an a high level of income. If a nation saves an invests only a small fraction of its income, its steaystate capital an income will be low. Let s now look at some ata to see if this theoretical result in fact helps explain the large international variation in stanars of living. Figure 76 is a scatterplot of ata from 96 countries. (The figure inclues most of the worl s economies. It exclues major oilproucing countries an countries that were communist uring much of this perio, because their experiences are explaine by their spe FIGURE 76 Income per person in 2003 (logarithmic scale) 100,000 10,000 1,000 Rwana Ethiopia Barbaos Cameroon El Salvaor Inia Nigeria Buruni Unite States Unite Kingom Argentina South Africa Togo Ghana Pakistan GuineaBissau Luxembourg Spain Greece Mexico Peru Ecuaor Switzerlan China Norway Japan Finlan South Korea Republic of Congo Thailan Zambia Investment as percentage of output (average ) International Evience on Investment Rates an Income per Person This scatterplot shows the experience of 96 countries, each represente by a single point. The horizontal axis shows the country s rate of investment, an the vertical axis shows the country s income per person. High investment is associate with high income per person, as the Solow moel preicts. Source: Alan Heston, Robert Summers, an Bettina Aten, Penn Worl Table Version 6.2, Center for International Comparisons of Prouction, Income an Prices at the University of Pennsylvania, September 2006.
13 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 203 cial circumstances.) The ata show a positive relationship between the fraction of output evote to investment an the level of income per person. That is, countries with high rates of investment, such as the Unite States an Japan, usually have high incomes, whereas countries with low rates of investment, such as Ethiopia an Buruni, have low incomes. Thus, the ata are consistent with the Solow moel s preiction that the investment rate is a key eterminant of whether a country is rich or poor. The strong correlation shown in this figure is an important fact, but it raises as many questions as it resolves. One might naturally ask, why o rates of saving an investment vary so much from country to country? There are many potential answers, such as tax policy, retirement patterns, the evelopment of financial markets, an cultural ifferences. In aition, political stability may play a role: not surprisingly, rates of saving an investment ten to be low in countries with frequent wars, revolutions, an coups. Saving an investment also ten to be low in countries with poor political institutions, as measure by estimates of official corruption. A final interpretation of the evience in Figure 76 is reverse causation: perhaps high levels of income somehow foster high rates of saving an investment. Unfortunately, there is no consensus among economists about which of the many possible explanations is most important. The association between investment rates an income per person is strong, an it is an important clue to why some countries are rich an others poor, but it is not the whole story. The correlation between these two variables is far from perfect. The Unite States an Peru, for instance, have ha similar investment rates, but income per person is more than eight times higher in the Unite States. There must be other eterminants of living stanars beyon saving an investment. Later in this chapter an also in the next one, we return to the international ifferences in income per person to see what other variables enter the picture. 72 The Golen Rule Level of Capital So far, we have use the Solow moel to examine how an economy s rate of saving an investment etermines its steaystate levels of capital an income. This analysis might lea you to think that higher saving is always a goo thing because it always leas to greater income. Yet suppose a nation ha a saving rate of 100 percent. That woul lea to the largest possible capital stock an the largest possible income. But if all of this income is save an none is ever consume, what goo is it? This section uses the Solow moel to iscuss the optimal amount of capital accumulation from the stanpoint of economic wellbeing. In the next chapter, we iscuss how government policies influence a nation s saving rate. But first, in this section, we present the theory behin these policy ecisions.
14 204 PART III Growth Theory: The Economy in the Very Long Run Comparing Steay States To keep our analysis simple, let s assume that a policymaker can set the economy s saving rate at any level. By setting the saving rate, the policymaker etermines the economy s steay state. What steay state shoul the policymaker choose? The policymaker s goal is to maximize the wellbeing of the iniviuals who make up the society. Iniviuals themselves o not care about the amount of capital in the economy, or even the amount of output. They care about the amount of goos an services they can consume. Thus, a benevolent policymaker woul want to choose the steay state with the highest level of consumption. The steaystate value of k that maximizes consumption is calle the Golen Rule level of capital an is enote k* gol. 2 How can we tell whether an economy is at the Golen Rule level? To answer this question, we must first etermine steaystate consumption per worker. Then we can see which steay state provies the most consumption. To fin steaystate consumption per worker, we begin with the national income accounts ientity an rearrange it as y = c + i c = y i. Consumption is output minus investment. Because we want to fin steaystate consumption, we substitute steaystate values for output an investment. Steaystate output per worker is f(k*), where k* is the steaystate capital stock per worker. Furthermore, because the capital stock is not changing in the steay state, investment equals epreciation k*. Substituting f(k*) for y an k* for i, we can write steaystate consumption per worker as c* = f(k*) k*. Accoring to this equation, steaystate consumption is what s left of steaystate output after paying for steaystate epreciation. This equation shows that an increase in steaystate capital has two opposing effects on steaystate consumption. On the one han, more capital means more output. On the other han, more capital also means that more output must be use to replace capital that is wearing out. Figure 77 graphs steaystate output an steaystate epreciation as a function of the steaystate capital stock. Steaystate consumption is the gap between output an epreciation. This figure shows that there is one level of the capital stock the Golen Rule level k* gol that maximizes consumption. When comparing steay states, we must keep in min that higher levels of capital affect both output an epreciation. If the capital stock is below the 2 Emun Phelps, The Golen Rule of Accumulation: A Fable for Growthmen, American Economic Review 51 (September 1961):
15 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 205 FIGURE 77 Steaystate output an epreciation k* gol Below the Golen Rule steay state, increases in steaystate capital raise steaystate consumption. c* gol Above the Golen Rule steay state, increases in steaystate capital reuce steaystate consumption. Steaystate epreciation (an investment), k* Steaystate output, f(k*) Steaystate capital per worker, k* SteayState Consumption The economy s output is use for consumption or investment. In the steay state, investment equals epreciation. Therefore, steaystate consumption is the ifference between output f( k*) an epreciation k*. Steaystate consumption is maximize at the Golen Rule steay state. The Golen Rule capital stock is enote k* gol, an the Golen Rule level of consumption is enote c* gol. Golen Rule level, an increase in the capital stock raises output more than epreciation, so consumption rises. In this case, the prouction function is steeper than the k* line, so the gap between these two curves which equals consumption grows as k* rises. By contrast, if the capital stock is above the Golen Rule level, an increase in the capital stock reuces consumption, because the increase in output is smaller than the increase in epreciation. In this case, the prouction function is flatter than the k* line, so the gap between the curves consumption shrinks as k* rises. At the Golen Rule level of capital, the prouction function an the k* line have the same slope, an consumption is at its greatest level. We can now erive a simple conition that characterizes the Golen Rule level of capital. Recall that the slope of the prouction function is the marginal prouct of capital MPK. The slope of the k* line is. Because these two slopes are equal at k* gol, the Golen Rule is escribe by the equation MPK =. At the Golen Rule level of capital, the marginal prouct of capital equals the epreciation rate. To make the point somewhat ifferently, suppose that the economy starts at some steaystate capital stock k* an that the policymaker is consiering increasing the capital stock to k* + 1. The amount of extra output from this increase in capital woul be f(k* + 1) f(k*), the marginal prouct of capital MPK. The amount of extra epreciation from having 1 more unit of capital is
16 206 PART III Growth Theory: The Economy in the Very Long Run the epreciation rate. Thus, the net effect of this extra unit of capital on consumption is MPK. If MPK > 0, then increases in capital increase con sumption, so k* must be below the Golen Rule level. If MPK < 0, then increases in capital ecrease consumption, so k* must be above the Golen Rule level. Therefore, the following conition escribes the Golen Rule: MPK = 0. At the Golen Rule level of capital, the marginal prouct of capital net of epreciation (MPK ) equals zero. As we will see, a policymaker can use this conition to fin the Golen Rule capital stock for an economy. 3 Keep in min that the economy oes not automatically gravitate towar the Golen Rule steay state. If we want any particular steaystate capital stock, such as the Golen Rule, we nee a particular saving rate to support it. Figure 78 shows the steay state if the saving rate is set to prouce the Golen Rule level of capital. If the saving rate is higher than the one use in this figure, the FIGURE 78 Steaystate output, epreciation, an investment per worker k* f(k*) c* gol s gol f(k*) i* gol k* gol Steaystate capital per worker, k* 1. To reach the Golen Rule steay state the economy nees the right saving rate. The Saving Rate an the Golen Rule There is only one saving rate that prouces the Golen Rule level of capital k* gol. Any change in the saving rate woul shift the sf(k) curve an woul move the economy to a steay state with a lower level of consumption. 3 Mathematical note: Another way to erive the conition for the Golen Rule uses a bit of calculus. Recall that c* = f(k*) k*. To fin the k* that maximizes c*, ifferentiate to fin c*/k* = f (k*) an set this erivative equal to zero. Noting that f (k*) is the marginal prouct of capital, we obtain the Golen Rule conition in the text.
17 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 207 steaystate capital stock will be too high. If the saving rate is lower, the steaystate capital stock will be too low. In either case, steaystate consumption will be lower than it is at the Golen Rule steay state. Fining the Golen Rule Steay State: A Numerical Example Consier the ecision of a policymaker choosing a steay state in the following economy. The prouction function is the same as in our earlier example: y = k. Output per worker is the square root of capital per worker. Depreciation is again 10 percent of capital. This time, the policymaker chooses the saving rate s an thus the economy s steay state. To see the outcomes available to the policymaker, recall that the following equation hols in the steay state: k* = s. f(k*) In this economy, this equation becomes k* s =. k* 0.1 Squaring both sies of this equation yiels a solution for the steaystate capital stock. We fin k* = 100s 2. Using this result, we can compute the steaystate capital stock for any saving rate. Table 73 presents calculations showing the steay states that result from various saving rates in this economy. We see that higher saving leas to a higher capital stock, which in turn leas to higher output an higher epreciation. Steaystate consumption, the ifference between output an epreciation, first rises with higher saving rates an then eclines. Consumption is highest when the saving rate is 0.5. Hence, a saving rate of 0.5 prouces the Golen Rule steay state. Recall that another way to ientify the Golen Rule steay state is to fin the capital stock at which the net marginal prouct of capital (MPK ) equals zero. For this prouction function, the marginal prouct is 4 1 MPK =. 2 k 4 Mathematical note: To erive this formula, note that the marginal prouct of capital is the erivative of the prouction function with respect to k.
18 208 PART III Growth Theory: The Economy in the Very Long Run TABLE 73 Fining the Golen Rule Steay State: A Numerical Example Assumptions: y = k ; = 0.1 s k* y* k* c* MPK MPK Using this formula, the last two columns of Table 73 present the values of MPK an MPK in the ifferent steay states. Note that the net marginal prouct of capital is exactly zero when the saving rate is at its Golen Rule value of 0.5. Because of iminishing marginal prouct, the net marginal prouct of capital is greater than zero whenever the economy saves less than this amount, an it is less than zero whenever the economy saves more. This numerical example confirms that the two ways of fining the Golen Rule steay state looking at steaystate consumption or looking at the marginal prouct of capital give the same answer. If we want to know whether an actual economy is currently at, above, or below its Golen Rule capital stock, the secon metho is usually more convenient, because it is relatively straightforwar to estimate the marginal prouct of capital. By contrast, evaluating an economy with the first metho requires estimates of steaystate consumption at many ifferent saving rates; such information is harer to obtain. Thus, when we apply this kin of analysis to the U.S. economy in the next chapter, we will evaluate U.S. saving by examining the marginal prouct of capital. Before engaging in that policy analysis, however, we nee to procee further in our evelopment an unerstaning of the Solow moel. The Transition to the Golen Rule Steay State Let s now make our policymaker s problem more realistic. So far, we have been assuming that the policymaker can simply choose the economy s steay state an jump there immeiately. In this case, the policymaker woul choose the steay state with highest consumption the Golen Rule steay state. But now suppose that the economy has reache a steay state other than the Golen Rule. What
19 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 209 happens to consumption, investment, an capital when the economy makes the transition between steay states? Might the impact of the transition eter the policymaker from trying to achieve the Golen Rule? We must consier two cases: the economy might begin with more capital than in the Golen Rule steay state, or with less. It turns out that the two cases offer very ifferent problems for policymakers. (As we will see in the next chapter, the secon case too little capital escribes most actual economies, incluing that of the Unite States.) Starting With Too Much Capital We first consier the case in which the economy begins at a steay state with more capital than it woul have in the Golen Rule steay state. In this case, the policymaker shoul pursue policies aime at reucing the rate of saving in orer to reuce the capital stock. Suppose that these policies succee an that at some point call it time t 0 the saving rate falls to the level that will eventually lea to the Golen Rule steay state. Figure 79 shows what happens to output, consumption, an investment when the saving rate falls. The reuction in the saving rate causes an immeiate increase in consumption an a ecrease in investment. Because investment an epreciation were equal in the initial steay state, investment will now be less than epreciation, which means the economy is no longer in a steay state. Graually, the capital stock falls, leaing to reuctions in output, consumption, an investment. These variables continue to fall until the economy reaches the new steay state. Because we are assuming that the new steay state is the Golen Rule steay state, consumption must be higher than it was before the change in the saving rate, even though output an investment are lower. FIGURE 79 Output, y Consumption, c Investment, i Reucing Saving When Starting With More Capital Than in the Golen Rule Steay State This figure shows what happens over time to output, consumption, an investment when the economy begins with more capital than the Golen Rule level an the saving rate is reuce. The reuction in the saving rate (at time t 0 ) causes an immeiate increase in consumption an an equal ecrease in investment. Over time, as the capital stock falls, output, consumption, an investment fall together. Because the economy began with too much capital, the new steay state has a higher level of consumption than the initial steay state. t 0 Time The saving rate is reuce.
20 210 PART III Growth Theory: The Economy in the Very Long Run Note that, compare to the ol steay state, consumption is higher not only in the new steay state but also along the entire path to it. When the capital stock excees the Golen Rule level, reucing saving is clearly a goo policy, for it increases consumption at every point in time. Starting With Too Little Capital When the economy begins with less capital than in the Golen Rule steay state, the policymaker must raise the saving rate to reach the Golen Rule. Figure 710 shows what happens. The increase in the saving rate at time t 0 causes an immeiate fall in consumption an a rise in investment. Over time, higher investment causes the capital stock to rise. As capital accumulates, output, consumption, an investment graually increase, eventually approaching the new steaystate levels. Because the initial steay state was below the Golen Rule, the increase in saving eventually leas to a higher level of consumption than that which prevaile initially. Does the increase in saving that leas to the Golen Rule steay state raise economic welfare? Eventually it oes, because the new steaystate level of consumption is higher than the initial level. But achieving that new steay state requires an initial perio of reuce consumption. Note the contrast to the case in which the economy begins above the Golen Rule. When the economy begins above the Golen Rule, reaching the Golen Rule prouces higher consumption at all points in time. When the economy begins below the Golen Rule, reaching the Golen Rule requires initially reucing consumption to increase consumption in the future. When eciing whether to try to reach the Golen Rule steay state, policymakers have to take into account that current consumers an future consumers are not always the same people. Reaching the Golen Rule achieves the highest steaystate level of consumption an thus benefits future generations. But when FIGURE 710 Output, y Consumption, c Investment, i t 0 Time Increasing Saving When Starting With Less Capital Than in the Golen Rule Steay State This figure shows what happens over time to output, consumption, an investment when the economy begins with less capital than the Golen Rule level an the saving rate is increase. The increase in the saving rate (at time t 0 ) causes an immeiate rop in consumption an an equal jump in investment. Over time, as the capital stock grows, output, consumption, an investment increase together. Because the economy began with less capital than the Golen Rule level, the new steay state has a higher level of consumption than the initial steay state. The saving rate is increase.
21 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 211 the economy is initially below the Golen Rule, reaching the Golen Rule requires raising investment an thus lowering the consumption of current generations. Thus, when choosing whether to increase capital accumulation, the policymaker faces a traeoff among the welfare of ifferent generations. A policymaker who cares more about current generations than about future ones may ecie not to pursue policies to reach the Golen Rule steay state. By contrast, a policymaker who cares about all generations equally will choose to reach the Golen Rule. Even though current generations will consume less, an infinite number of future generations will benefit by moving to the Golen Rule. Thus, optimal capital accumulation epens crucially on how we weigh the interests of current an future generations. The biblical Golen Rule tells us, o unto others as you woul have them o unto you. If we hee this avice, we give all generations equal weight. In this case, it is optimal to reach the Golen Rule level of capital which is why it is calle the Golen Rule. 73 Population Growth The basic Solow moel shows that capital accumulation, by itself, cannot explain sustaine economic growth: high rates of saving lea to high growth temporarily, but the economy eventually approaches a steay state in which capital an output are constant. To explain the sustaine economic growth that we observe in most parts of the worl, we must expan the Solow moel to incorporate the other two sources of economic growth population growth an technological progress. In this section we a population growth to the moel. Instea of assuming that the population is fixe, as we i in Sections 71 an 72, we now suppose that the population an the labor force grow at a constant rate n. For example, the U.S. population grows about 1 percent per year, so n = This means that if 150 million people are working one year, then million ( ) are working the next year, an million ( ) the year after that, an so on. The Steay State With Population Growth How oes population growth affect the steay state? To answer this question, we must iscuss how population growth, along with investment an epreciation, influences the accumulation of capital per worker. As we note before, investment raises the capital stock, an epreciation reuces it. But now there is a thir force acting to change the amount of capital per worker: the growth in the number of workers causes capital per worker to fall. We continue to let lowercase letters stan for quantities per worker. Thus, k = K/L is capital per worker, an y = Y/L is output per worker. Keep in min, however, that the number of workers is growing over time. The change in the capital stock per worker is D k = i ( + n)k.
22 212 PART III Growth Theory: The Economy in the Very Long Run This equation shows how investment, epreciation, an population growth influence the perworker capital stock. Investment increases k, whereas epreciation an population growth ecrease k. We saw this equation earlier in this chapter for the special case of a constant population (n = 0). We can think of the term ( + n)k as efining breakeven investment the amount of investment necessary to keep the capital stock per worker constant. Breakeven investment inclues the epreciation of existing capital, which equals k. It also inclues the amount of investment necessary to provie new workers with capital. The amount of investment necessary for this purpose is nk, because there are n new workers for each existing worker an because k is the amount of capital for each worker. The equation shows that population growth reuces the accumulation of capital per worker much the way epreciation oes. Depreciation reuces k by wearing out the capital stock, whereas population growth reuces k by spreaing the capital stock more thinly among a larger population of workers. 5 Our analysis with population growth now procees much as it i previously. First, we substitute sf(k) for i. The equation can then be written as D k = sf(k) ( + n)k. To see what etermines the steaystate level of capital per worker, we use Figure 711, which extens the analysis of Figure 74 to inclue the effects of pop FIGURE 711 Investment, breakeven investment Breakeven investment, ( + n)k Investment, sf(k) k* Capital per worker, k The steay state Population Growth in the Solow Moel Depreciation an population growth are two reasons the capital stock per worker shrinks. If n is the rate of population growth an is the rate of epreciation, then ( + n)k is breakeven investment the amount of investment necessary to keep constant the capital stock per worker k. For the economy to be in a steay state, investment sf(k) must offset the effects of epreciation an population growth ( + n)k. This is represente by the crossing of the two curves. 5 Mathematical note: Formally eriving the equation for the change in k requires a bit of calculus. Note that the change in k per unit of time is k/t = (K/L)/t. After applying the stanar rules of calculus, we can write this as k/t = (1/L)(K/t) (K/L 2 )(L/t). Now use the following facts to substitute in this equation: K/t = I K an (L/t)/L = n. After a bit of manipulation, this prouces the equation in the text.
23 CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth 213 ulation growth. An economy is in a steay state if capital per worker k is unchanging. As before, we esignate the steaystate value of k as k*. If k is less than k*, investment is greater than breakeven investment, so k rises. If k is greater than k*, investment is less than breakeven investment, so k falls. In the steay state, the positive effect of investment on the capital stock per worker exactly balances the negative effects of epreciation an population growth. That is, at k*, D k = 0 an i* = k* + nk*. Once the economy is in the steay state, investment has two purposes. Some of it ( k*) replaces the epreciate capital, an the rest (nk*) provies the new workers with the steaystate amount of capital. The Effects of Population Growth Population growth alters the basic Solow moel in three ways. First, it brings us closer to explaining sustaine economic growth. In the steay state with population growth, capital per worker an output per worker are constant. Because the number of workers is growing at rate n, however, total capital an total output must also be growing at rate n. Hence, although population growth cannot explain sustaine growth in the stanar of living (because output per worker is constant in the steay state), it can help explain sustaine growth in total output. Secon, population growth gives us another explanation for why some countries are rich an others are poor. Consier the effects of an increase in population growth. Figure 712 shows that an increase in the rate of population FIGURE 712 Investment, breakeven investment 1. An increase in the rate of population growth... ( + n 2 )k ( + n 1 )k sf(k) The Impact of Population Growth An increase in the rate of population growth from n 1 to n 2 shifts the line representing population growth an epreciation upwar. The new steay state k 2 * has a lower level of capital per worker than the initial steay state k 1 *. Thus, the Solow moel preicts that economies with higher rates of population growth will have lower levels of capital per worker an therefore lower incomes. k* 2 k* 1 Capital per worker, k reuces the steaystate capital stock.
Economic Growth I: Capital Accumulation and Population Growth
CHAPTER 8 : Capital Accumulation and Population Growth Modified for ECON 2204 by Bob Murphy 2016 Worth Publishers, all rights reserved IN THIS CHAPTER, YOU WILL LEARN: the closed economy Solow model how
More informationCHAPTER 7 Economic Growth I
CHAPTER 7 Economic Growth I Questions for Review 1. In the Solow growth model, a high saving rate leads to a large steadystate capital stock and a high level of steadystate output. A low saving rate
More informationModules 6 and 7: Markets, Prices, Supply, and Demand practice problems. Practice problems and illustrative test questions for the final exam
Moules 6 an 7: Markets, Prices, Supply, an Deman practice problems Practice problems an illustrative test questions for the final exam (The attache PDF file has better formatting.) This posting gives sample
More informationReview Questions  CHAPTER 8
Review Questions  CHAPTER 8 1. The formula for steadystate consumption per worker (c*) as a function of output per worker and investment per worker is: A) c* = f(k*) δk*. B) c* = f(k*) + δk*. C) c* =
More informationMathematics Review for Economists
Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school
More informationAn intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations
This page may be remove to conceal the ientities of the authors An intertemporal moel of the real exchange rate, stock market, an international ebt ynamics: policy simulations Saziye Gazioglu an W. Davi
More informationCHAPTER 5 : CALCULUS
Dr Roger Ni (Queen Mary, University of Lonon)  5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective
More information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationChapter 4: Elasticity
Chapter : Elasticity Elasticity of eman: It measures the responsiveness of quantity emane (or eman) with respect to changes in its own price (or income or the price of some other commoity). Why is Elasticity
More informationAPPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS
Application of Calculus in Commerce an Economics 41 APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS æ We have learnt in calculus that when 'y' is a function of '', the erivative of y w.r.to i.e. y ö
More informationChapters 7 and 8 Solow Growth Model Basics
Chapters 7 and 8 Solow Growth Model Basics The Solow growth model breaks the growth of economies down into basics. It starts with our production function Y = F (K, L) and puts in perworker terms. Y L
More informationNotes on the Theories of Growth
Notes on the Theories of Growth Economic Growth & Development These recitation notes cover basic concepts in economic growth theory relevant to the cases seen in class (Singapore, Indonesia, Japan, and
More informationPurpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of
Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose
More informationState of Louisiana Office of Information Technology. Change Management Plan
State of Louisiana Office of Information Technology Change Management Plan Table of Contents Change Management Overview Change Management Plan Key Consierations Organizational Transition Stages Change
More informationThere are two different ways you can interpret the information given in a demand curve.
Econ 500 Microeconomic Review Deman What these notes hope to o is to o a quick review of supply, eman, an equilibrium, with an emphasis on a more quantifiable approach. Deman Curve (Big icture) The whole
More informationData Center Power System Reliability Beyond the 9 s: A Practical Approach
Data Center Power System Reliability Beyon the 9 s: A Practical Approach Bill Brown, P.E., Square D Critical Power Competency Center. Abstract Reliability has always been the focus of missioncritical
More informationMASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT
MASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT ECONOMICS OF GROWTH AND INNOVATION Lecture 1, January 23, 2004 Theories of Economic Growth 1. Introduction 2. Exogenous Growth The Solow Model Mandatory
More informationOptimal Control Policy of a Production and Inventory System for multiproduct in Segmented Market
RATIO MATHEMATICA 25 (2013), 29 46 ISSN:15927415 Optimal Control Policy of a Prouction an Inventory System for multiprouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational
More information9.3. Diffraction and Interference of Water Waves
Diffraction an Interference of Water Waves 9.3 Have you ever notice how people relaxing at the seashore spen so much of their time watching the ocean waves moving over the water, as they break repeately
More informationStudy on the Price Elasticity of Demand of Beijing Subway
Journal of Traffic an Logistics Engineering, Vol, 1, No. 1 June 2013 Stuy on the Price Elasticity of Deman of Beijing Subway Yanan Miao an Liang Gao MOE Key Laboratory for Urban Transportation Complex
More informationEconomic Growth. (c) Copyright 1999 by Douglas H. Joines 1
Economic Growth (c) Copyright 1999 by Douglas H. Joines 1 Module Objectives Know what determines the growth rates of aggregate and per capita GDP Distinguish factors that affect the economy s growth rate
More informationThe Inefficiency of Marginal cost pricing on roads
The Inefficiency of Marginal cost pricing on roas Sofia GrahnVoornevel Sweish National Roa an Transport Research Institute VTI CTS Working Paper 4:6 stract The economic principle of roa pricing is that
More informationModelling and Resolving Software Dependencies
June 15, 2005 Abstract Many Linux istributions an other moern operating systems feature the explicit eclaration of (often complex) epenency relationships between the pieces of software
More informationProfessional Level Options Module, Paper P4(SGP)
Answers Professional Level Options Moule, Paper P4(SGP) Avance Financial Management (Singapore) December 2007 Answers Tutorial note: These moel answers are consierably longer an more etaile than woul be
More informationEnterprise Resource Planning
Enterprise Resource Planning MPC 6 th Eition Chapter 1a McGrawHill/Irwin Copyright 2011 by The McGrawHill Companies, Inc. All rights reserve. Enterprise Resource Planning A comprehensive software approach
More informationName: Date: 3. Variables that a model tries to explain are called: A. endogenous. B. exogenous. C. market clearing. D. fixed.
Name: Date: 1 A measure of how fast prices are rising is called the: A growth rate of real GDP B inflation rate C unemployment rate D marketclearing rate 2 Compared with a recession, real GDP during a
More informationFirewall Design: Consistency, Completeness, and Compactness
C IS COS YS TE MS Firewall Design: Consistency, Completeness, an Compactness Mohame G. Goua an XiangYang Alex Liu Department of Computer Sciences The University of Texas at Austin Austin, Texas 787121188,
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationTopics in Macroeconomics 2 Econ 2004
Topics in Macroeconomics 2 Econ 2004 Practice Questions for Master Class 2 on Monay, March 2n, 2009 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationOutline of model. Factors of production 1/23/2013. The production function: Y = F(K,L) ECON 3010 Intermediate Macroeconomics
ECON 3010 Intermediate Macroeconomics Chapter 3 National Income: Where It Comes From and Where It Goes Outline of model A closed economy, marketclearing model Supply side factors of production determination
More informationMandateBased Health Reform and the Labor Market: Evidence from the Massachusetts Reform
ManateBase Health Reform an the Labor Market: Evience from the Massachusetts Reform Jonathan T. Kolsta Wharton School, University of Pennsylvania an NBER Amana E. Kowalski Department of Economics, Yale
More informationChapter 7: Economic Growth part 1
Chapter 7: Economic Growth part 1 Learn the closed economy Solow model See how a country s standard of living depends on its saving and population growth rates Learn how to use the Golden Rule to find
More informationIntroduction to Integration Part 1: AntiDifferentiation
Mathematics Learning Centre Introuction to Integration Part : AntiDifferentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
More informationProduct Differentiation for SoftwareasaService Providers
University of Augsburg Prof. Dr. Hans Ulrich Buhl Research Center Finance & Information Management Department of Information Systems Engineering & Financial Management Discussion Paper WI99 Prouct Differentiation
More informationCURRENCY OPTION PRICING II
Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility BlackScholes Moel for Currency Options Properties of the BS Moel Option Sensitivity
More informationJON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT
OPTIMAL INSURANCE COVERAGE UNDER BONUSMALUS CONTRACTS BY JON HOLTAN if P&C Insurance Lt., Oslo, Norway ABSTRACT The paper analyses the questions: Shoul or shoul not an iniviual buy insurance? An if so,
More informationDetecting Possibly Fraudulent or ErrorProne Survey Data Using Benford s Law
Detecting Possibly Frauulent or ErrorProne Survey Data Using Benfor s Law Davi Swanson, Moon Jung Cho, John Eltinge U.S. Bureau of Labor Statistics 2 Massachusetts Ave., NE, Room 3650, Washington, DC
More informationf(x) = a x, h(5) = ( 1) 5 1 = 2 2 1
Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)
More informationAnswers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More informationNotes on tangents to parabolas
Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,
More informationMeasures of distance between samples: Euclidean
4 Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate
More informationShortRun Production and Costs
ShortRun Production and Costs The purpose of this section is to discuss the underlying work of firms in the shortrun the production of goods and services. Why is understanding production important to
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying
More informationJitter effects on Analog to Digital and Digital to Analog Converters
Jitter effects on Analog to Digital an Digital to Analog Converters Jitter effects copyright 1999, 2000 Troisi Design Limite Jitter One of the significant problems in igital auio is clock jitter an its
More informationDebt cycles, instability and fiscal rules: a GodleyMinsky model
Faculty of usiness an Law Debt cycles, instability an fiscal rules: a GoleyMinsky moel Yannis Dafermos Department of Accounting, Economics an Finance, University of the West of Englan, ristol, UK Yannis.Dafermos@uwe.ac.uk
More informationA Generalization of Sauer s Lemma to Classes of LargeMargin Functions
A Generalization of Sauer s Lemma to Classes of LargeMargin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/
More informationChapter 3. The Concept of Elasticity and Consumer and Producer Surplus. Chapter Objectives. Chapter Outline
Chapter 3 The Concept of Elasticity and Consumer and roducer Surplus Chapter Objectives After reading this chapter you should be able to Understand that elasticity, the responsiveness of quantity to changes
More informationRules for Finding Derivatives
3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to
More informationHull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5
Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. OnePerio Binomial Moel Creating synthetic options (replicating options)
More informationy or f (x) to determine their nature.
Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic
More informationRisk Management for Derivatives
Risk Management or Derivatives he Greeks are coming the Greeks are coming! Managing risk is important to a large number o iniviuals an institutions he most unamental aspect o business is a process where
More informationChapter 4 Online Appendix: The Mathematics of Utility Functions
Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can
More informationWeb Appendices of Selling to Overcon dent Consumers
Web Appenices of Selling to Overcon ent Consumers Michael D. Grubb A Option Pricing Intuition This appenix provies aitional intuition base on option pricing for the result in Proposition 2. Consier the
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms
More informationChapter 6 Competitive Markets
Chapter 6 Competitive Markets After reading Chapter 6, COMPETITIVE MARKETS, you should be able to: List and explain the characteristics of Perfect Competition and Monopolistic Competition Explain why a
More informationReading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14.
7 Shocks Reaing: Ryen chs 3 & 4, Shu chs 5 & 6 For the enthusiasts, Shu chs 3 & 4 A goo article for further reaing: Shull & Draine, The physics of interstellar shock waves, in Interstellar processes; Proceeings
More informationExponential Functions: Differentiation and Integration. The Natural Exponential Function
46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More informationCalculus Refresher, version 2008.4. c 19972008, Paul Garrett, garrett@math.umn.edu http://www.math.umn.edu/ garrett/
Calculus Refresher, version 2008.4 c 9972008, Paul Garrett, garrett@math.umn.eu http://www.math.umn.eu/ garrett/ Contents () Introuction (2) Inequalities (3) Domain of functions (4) Lines (an other items
More informationOption Pricing for Inventory Management and Control
Option Pricing for Inventory Management an Control Bryant Angelos, McKay Heasley, an Jeffrey Humpherys Abstract We explore the use of option contracts as a means of managing an controlling inventories
More informationNote on growth and growth accounting
CHAPTER 0 Note on growth and growth accounting 1. Growth and the growth rate In this section aspects of the mathematical concept of the rate of growth used in growth models and in the empirical analysis
More informationExample Optimization Problems selected from Section 4.7
Example Optimization Problems selecte from Section 4.7 19) We are aske to fin the points ( X, Y ) on the ellipse 4x 2 + y 2 = 4 that are farthest away from the point ( 1, 0 ) ; as it happens, this point
More informationTO DETERMINE THE SHELF LIFE OF IBUPROFEN SOLUTION
TO DETERMINE THE SHELF LIFE OF IBUPROFEN SOLUTION AIM: To etermine the shelf life of solution using accelerate stability stuies. MATERIAL REQUIREMENT, ethanol, phenolphthalein, soium hyroxie (0.1N), glass
More informationSearch Advertising Based Promotion Strategies for Online Retailers
Search Avertising Base Promotion Strategies for Online Retailers Amit Mehra The Inian School of Business yeraba, Inia Amit Mehra@isb.eu ABSTRACT Web site aresses of small on line retailers are often unknown
More informationThe oneyear nonlife insurance risk
The oneyear nonlife insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on nonlife insurance reserve risk has been evote to the ultimo risk, the risk in the
More informationINFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES
1 st Logistics International Conference Belgrae, Serbia 2830 November 2013 INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES Goran N. Raoičić * University of Niš, Faculty of Mechanical
More informationLecture L253D Rigid Body Kinematics
J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L253D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general threeimensional
More informationTRADE AND INVESTMENT IN THE NATIONAL ACCOUNTS This text accompanies the material covered in class.
TRADE AND INVESTMENT IN THE NATIONAL ACCOUNTS This text accompanies the material covered in class. 1 Definition of some core variables Imports (flow): Q t Exports (flow): X t Net exports (or Trade balance)
More informationStock Market Value Prediction Using Neural Networks
Stock Market Value Preiction Using Neural Networks Mahi Pakaman Naeini IT & Computer Engineering Department Islamic Aza University Paran Branch email: m.pakaman@ece.ut.ac.ir Hamireza Taremian Engineering
More informationDIFFRACTION AND INTERFERENCE
DIFFRACTION AND INTERFERENCE In this experiment you will emonstrate the wave nature of light by investigating how it bens aroun eges an how it interferes constructively an estructively. You will observe
More informationInnovation Union means: More jobs, improved lives, better society
The project follows the Lisbon an Gothenburg Agenas, an supports the EU 2020 Strategy, in particular SMART Growth an the Innovation Union: Innovation Union means: More jobs, improve lives, better society
More informationDEMAND AND SUPPLY IN FACTOR MARKETS
Chapter 14 DEMAND AND SUPPLY IN FACTOR MARKETS Key Concepts Prices and Incomes in Competitive Factor Markets Factors of production (labor, capital, land, and entrepreneurship) are used to produce output.
More informationMODELLING OF TWO STRATEGIES IN INVENTORY CONTROL SYSTEM WITH RANDOM LEAD TIME AND DEMAND
art I. robobabilystic Moels Computer Moelling an New echnologies 27 Vol. No. 23 ransport an elecommunication Institute omonosova iga V9 atvia MOEING OF WO AEGIE IN INVENOY CONO YEM WIH ANOM EA IME AN
More informationMSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUMLIKELIHOOD ESTIMATION
MAXIMUMLIKELIHOOD ESTIMATION The General Theory of ML Estimation In orer to erive an ML estimator, we are boun to make an assumption about the functional form of the istribution which generates the
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationISSN: 22773754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014
ISSN: 77754 ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Volume, Issue, June 04 Manufacturing process with isruption uner Quaratic Deman for Deteriorating Inventory
More informationEC2105, Professor Laury EXAM 2, FORM A (3/13/02)
EC2105, Professor Laury EXAM 2, FORM A (3/13/02) Print Your Name: ID Number: Multiple Choice (32 questions, 2.5 points each; 80 points total). Clearly indicate (by circling) the ONE BEST response to each
More informationConsumer Referrals. Maria Arbatskaya and Hideo Konishi. October 28, 2014
Consumer Referrals Maria Arbatskaya an Hieo Konishi October 28, 2014 Abstract In many inustries, rms rewar their customers for making referrals. We analyze the optimal policy mix of price, avertising intensity,
More informationRisk Adjustment for Poker Players
Risk Ajustment for Poker Players William Chin DePaul University, Chicago, Illinois Marc Ingenoso Conger Asset Management LLC, Chicago, Illinois September, 2006 Introuction In this article we consier risk
More information7 AGGREGATE SUPPLY AND AGGREGATE DEMAND* Chapter. Key Concepts
Chapter 7 AGGREGATE SUPPLY AND AGGREGATE DEMAND* Key Concepts Aggregate Supply The aggregate production function shows that the quantity of real GDP (Y ) supplied depends on the quantity of labor (L ),
More information20122013 Enhanced Instructional Transition Guide Mathematics Algebra I Unit 08
01013 Enhance Instructional Transition Guie Unit 08: Exponents an Polynomial Operations (18 ays) Possible Lesson 01 (4 ays) Possible Lesson 0 (7 ays) Possible Lesson 03 (7 ays) POSSIBLE LESSON 0 (7 ays)
More informationCALCULATION INSTRUCTIONS
Energy Saving Guarantee Contract ppenix 8 CLCULTION INSTRUCTIONS Calculation Instructions for the Determination of the Energy Costs aseline, the nnual mounts of Savings an the Remuneration 1 asics ll prices
More informationKater Pendulum. Introduction. It is wellknown result that the period T of a simple pendulum is given by. T = 2π
Kater Penulum ntrouction t is wellknown result that the perio of a simple penulum is given by π L g where L is the length. n principle, then, a penulum coul be use to measure g, the acceleration of gravity.
More informationCHAPTER 7: AGGREGATE DEMAND AND AGGREGATE SUPPLY
CHAPTER 7: AGGREGATE DEMAND AND AGGREGATE SUPPLY Learning goals of this chapter: What forces bring persistent and rapid expansion of real GDP? What causes inflation? Why do we have business cycles? How
More informationAverageCase Analysis of a Nearest Neighbor Algorithm. Moett Field, CA USA. in accuracy. neighbor.
From Proceeings of the Thirteenth International Joint Conference on Articial Intelligence (1993). Chambery, France: Morgan Kaufmann AverageCase Analysis of a Nearest Neighbor Algorithm Pat Langley Learning
More informationDETERMINING OPTIMAL STOCK LEVEL IN MULTIECHELON SUPPLY CHAINS
HUNGARIAN JOURNA OF INDUSTRIA CHEMISTRY VESZPRÉM Vol. 39(1) pp. 107112 (2011) DETERMINING OPTIMA STOCK EVE IN MUTIECHEON SUPPY CHAINS A. KIRÁY 1, G. BEVÁRDI 2, J. ABONYI 1 1 University of Pannonia, Department
More informationSustainability Through the Market: Making Markets Work for Everyone q
www.corporateenvstrategy.com Sustainability an the Market Sustainability Through the Market: Making Markets Work for Everyone q Peter White Sustainable evelopment is about ensuring a better quality of
More informationLagrange Multipliers without Permanent Scarring
Lagrange Multipliers without Permanent Scarring Dan Klein Introuction This tutorial assumes that you want to know what Lagrange multipliers are, but are more intereste in getting the intuitions an central
More informationDifferentiability of Exponential Functions
Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an
More informationDigital barrier option contract with exponential random time
IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun
More informationINTRODUCTION TO BEAMS
CHAPTER Structural Steel Design LRFD etho INTRODUCTION TO BEAS Thir Eition A. J. Clark School of Engineering Department of Civil an Environmental Engineering Part II Structural Steel Design an Analsis
More informationOn Adaboost and Optimal Betting Strategies
On Aaboost an Optimal Betting Strategies Pasquale Malacaria 1 an Fabrizio Smerali 1 1 School of Electronic Engineering an Computer Science, Queen Mary University of Lonon, Lonon, UK Abstract We explore
More informationElasticity. I. What is Elasticity?
Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in
More informationMACROECONOMICS SECTION
MACROECONOMICS SECTION GENERAL TIPS Be sure every graph is carefully labeled and explained. Every answer must include a section that contains a response to WHY the result holds. Good resources include
More informationInternational Economic Relations
International Economic Relations Prof. Murphy Chapter 8 Krugman and Obstfeld 1. The import demand equation, MD, is found by subtracting the home supply equation from the home demand equation. This results
More informationFISCAL POLICY* Chapter. Key Concepts
Chapter 11 FISCAL POLICY* Key Concepts The Federal Budget The federal budget is an annual statement of the government s expenditures and tax revenues. Using the federal budget to achieve macroeconomic
More information( ) = ( ) = ( + ) which means that both capital and output grow permanently at a constant rate
1 Endogenous Growth We present two models that are very popular in the, socalled, new growth theory literature. They represent economies where, notwithstanding the absence of exogenous technical progress,
More informationCalculating Viscous Flow: Velocity Profiles in Rivers and Pipes
previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar
More informationAchieving quality audio testing for mobile phones
Test & Measurement Achieving quality auio testing for mobile phones The auio capabilities of a cellular hanset provie the funamental interface between the user an the raio transceiver. Just as RF testing
More information