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Economic Growth I

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The central equation of the Solow model first appears on slide 24. The textbook does not assign a special name to this equation, but the equation is referred to very often, so giving it a name makes life easier for students and for the instructor. In these slides, therefore, it’s referred to as “the equation of motion for k.”

Because this name is not used in the textbook, one could reasonably argue that it should not be used here. If you take this position, please let me know in an email (roncron@unlv.edu). I am doing annual updates to these slides (which will publish in summer 2003 and summer 2004), so I will have the chance to incorporate feedback from users like yourself into the revisions.

One final note before we begin: I strongly encourage you to take a look at William Easterly’s brilliant new book The Elusive Quest for Growth (MIT Press, 2001). It has lots of compelling real world examples (some of which I’ve included in the PowerPoint presentation of the two economic growth chapters). It also explains economic theory in ways that students (and college-educated non-econ majors) find easy to grasp. After reading it, I find I do a better job teaching Mankiw’s growth chapters to my students.

The following slides present selected poverty statistics and estimates of the reduction in poverty caused by economic growth.

The point: If we can learn how to help poor countries rise out of poverty, we will be making a huge difference in the lives of billions of people, as well as creating new markets for our exports.

…for rich countries:

This is just one of many concrete examples of the negative relationship between economic growth and the poverty rate.

Again, the point here is this:

Learning about economic growth

The following slides present selected poverty statistics and estimates of the reduction in poverty caused by economic growth.

The point: If we can learn how to help poor countries rise out of poverty, we will be making a huge difference in the lives of billions of people, as well as creating new markets for our exports.

…for rich countries:

In case you’re wondering how I did this calculation:

Computed actual quarterly growth rate of real income per capita from 1989:4 through 1999:4.

Added one-fourth of one-tenth of one percent to each quarter’s actual growth rate.

Computed what real income per capita would have been with the new growth rates.

Multiplied this hypothetical real income per capita by the population to get hypothetical real GDP.

Computed the difference between hypothetical and actual real GDP.

Cumulated these differences over the period 1990:1-1999:4.

Like the original real GDP data, the cumulative difference was in 1996 dollars. I multiplied this amount by 10%, the amount by which the GDP deflator rose between 1996 and 2002:1, so the final result ($449 billion) is expressed in 2002:1 dollars, which I simply call “today’s dollars.”

DATA SOURCE:

Real GDP, GDP deflator - Dept of Commerce, Bureau of Economic Analysis.

Population - Dept of Commerce, Census Bureau.

All obtained from “FRED” - the St. Louis Fed’s database, on the web at http://www.stls.frb.org/fred/.

f(k) is the “per worker production function,” it shows how much output one worker could produce using k units of capital.

You might want to point out that this is the same production function we worked with in chapter 3. We’re just expressing it differently.

I.e.,

“The economy saves three-tenths of income,”

“every year, 10% of the capital stock wears out,” and

“suppose the economy starts out with four units of capital for every worker.”

Then, ask them to determine the numbers for the second row and write them in their notes.

After the second round of this, it’s probably fine to just show them the rest of the table.

Also, this gives students a little psychological momentum to make it easier for them to start on the end-of-chapter exercises (if you assign them as homework).

(If any need a hint, remind them that the steady state is defined by k = 0. A further hint is that they answers they get should be the same as the last row of the big table on the preceding slide, since we are still using all the same parameter values.)

An earlier slide said that the model’s omission of G and T was only to simplify the presentation. We can still do policy analysis. We know from Chapter 3 that changes in G and/or T affect national saving. In the Solow model as presented here, we can simply change the exogenous saving rate to analyze the impact of fiscal policy changes.

In the static model of Chapter 3, we learned that a fiscal expansion crowds out investment. The Solow model allows us to see the long-run dynamic effects: the fiscal expansion, by reducing the saving rate, reduces investment. If we were initially in a steady state (in which investment just covers depreciation), then the fall in investment will cause capital per worker, labor productivity, and income per capita to fall toward a new, lower steady state. (If we were initially below a steady state, then the fiscal expansion causes capital per worker and productivity to grow more slowly, and reduces their steady-state values.)

Be sure to clarify the differences:

On this graph, the horizontal axis measures k*, not k. Thus, once we have found k* using the other graph, we plot that k* on this graph to see where the economy’s steady state is in relation to the golden rule capital stock.

On this graph, the curve measures f(k*), not sf(k).

On the other diagram, the intersection of the two curves determines k*. On this graph, the only thing determined by the intersection of the two curves is the level of capital where c*=0, and we certainly wouldn’t want to be there.

There are no dynamics in this graph, as we are in a steady state. In the other graph, the gap between the two curves determines the change in capital.

The problem is to find the value of k* that maximizes c* = f(k*) k*.

Just take the first derivative of that expression and set equal to zero:

f(k*) = 0

where f(k*) = MPK = slope of production function and = slope of steady-state investment line.

- changing G or T affects national saving

- holding T constant overall, but changing the structure of the tax system to provide more incentives for private saving (i.e., shifting from income tax to consumption tax in such a way that leaves total revenue unchanged)

Before t0: in a steady state, where k, y, c, and i are all constant.

At t0:

The change in the saving rate doesn’t immediately change k, so y doesn’t change immediately.

But the fall in s causes a fall in investment [because saving equals investment] and a rise in consumption [because c = (1-s)y, s has fallen but y has not yet changed.]. Note that c = -i, because y = c + i and y has not changed.

After t0:

In the previous steady state, saving and investment were just enough to cover depreciation. Then saving and investment were reduced, so depreciation is greater than investment, which causes k to fall toward a new, lower steady state value.

As k falls and settles on its new, lower steady state value, so will y, c, and i (because each of them is a function of k).

Even though c is falling, it doesn’t fall all the way back to its initial value.

Policymakers would be happy to make this change, as it produces higher consumption at all points in time (relative to what consumption would have been if the saving rate had not been reduced.

At t0:

The increase in s doesn’t immediately change k, so y doesn’t change immediately.

But the increase in s causes investment to rise [because higher saving means higher investment] and consumption to fall [because we are saving more of our income, and consuming less of it].

After t0:

Now, saving and investment exceed depreciation, so k starts rising toward a new, higher steady state value. The behavior of k causes the same behavior in y, c, and i (qualitatively the same, that is).

Ultimately, consumption ends up at a higher steady state level. But initially consumption falls. Therefore, if policymakers value the current generation’s well-being more than that of future generations, they might be reluctant to adjust the saving rate to achieve the Golden Rule. Notice, though, that if they did increase s, an infinite number of future generations would benefit, which makes the sacrifice of the current generation seem more acceptable.

In the static model of Chapter 3, we learned that a fiscal expansion crowds out investment. The Solow model allows us to see the long-run dynamic effects: the fiscal expansion, by reducing the saving rate, reduces investment. If we were initially in a steady state (in which investment just covers depreciation), then the fall in investment will cause capital per worker, labor productivity, and income per capita to fall toward a new, lower steady state. (If we were initially below a steady state, then the fiscal expansion causes capital per worker and productivity to grow more slowly, and reduces their steady-state values.)

Chapter 8 addresses this issue by introducing technological progress into the Solow model.

- 1. macroeconomics fifth edition N. Gregory Mankiw PowerPoint® Slides by Ron Cronovich CHAPTER SEVEN Economic Growth Imacro © 2002 Worth Publishers, all rights reserved
- 2. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 2 Chapter 7 learning objectivesChapter 7 learning objectives 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 the optimal savings rate and capital stock
- 3. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 3 The importance ofThe importance of economic growtheconomic growth The importance ofThe importance of economic growtheconomic growth …for poor countries
- 4. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 4 selected poverty statisticsselected poverty statistics In the poorest one-fifth of all countries, daily caloric intake is 1/3 lower than in the richest fifth the infant mortality rate is 200 per 1000 births, compared to 4 per 1000 births in the richest fifth.
- 5. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 5 selected poverty statisticsselected poverty statistics In Pakistan, 85% of people live on less than $2/day One-fourth of the poorest countries have had famines during the past 3 decades. (none of the richest countries had famines) Poverty is associated with the oppression of women and minorities
- 6. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 6 Estimated effects of economic growthEstimated effects of economic growth A 10% increase in income is associated with a 6% decrease in infant mortality Income growth also reduces poverty. Example: +65%-12%1997-99 -25%+76% Growth and Poverty in Indonesia 1984-96 change in # of persons living below poverty line change in income per capita
- 7. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 7 Income and poverty in the worldIncome and poverty in the world selected countries, 2000selected countries, 2000 0 10 20 30 40 50 60 70 80 90 100 $0 $5,000 $10,000 $15,000 $20,000 Income per capita in dollars %ofpopulation livingon$2perdayorless Madagascar India Bangladesh Nepal Botswana Mexico Chile S. Korea Brazil Russian Federation Thailand Peru China Kenya
- 8. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 8 The importance ofThe importance of economic growtheconomic growth The importance ofThe importance of economic growtheconomic growth …for poor countries …for rich countries
- 9. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 9 Huge effects from tiny differencesHuge effects from tiny differences In rich countries like the U.S., if government policies or “shocks” have even a small impact on the long-run growth rate, they will have a huge impact on our standard of living in the long run…
- 10. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 10 Huge effects from tiny differencesHuge effects from tiny differences 1,081.4%243.7%85.4% 624.5%169.2%64.0% 2.5% 2.0% …100 years…50 years…25 years percentage increase in standard of living after… annual growth rate of income per capita
- 11. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 11 Huge effects from tiny differencesHuge effects from tiny differences If the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher during the 1990s, the U.S. would have generated an additional $449 billion of income during that decade
- 12. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 12 The lessons of growth theoryThe lessons of growth theory ……can make a positive difference in thecan make a positive difference in the lives of hundreds of millions of people.lives of hundreds of millions of people. These lessons help us understand why poor countries are poor design policies that can help them grow learn how our own growth rate is affected by shocks and our government’s policies
- 13. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 13 The Solow ModelThe Solow Model due to Robert Solow, won Nobel Prize for contributions to the study of economic growth a major paradigm: – widely used in policy making – benchmark against which most recent growth theories are compared looks at the determinants of economic growth and the standard of living in the long run
- 14. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 14 How Solow model is different fromHow Solow model is different from Chapter 3’s modelChapter 3’s model 1. K is no longer fixed: investment causes it to grow, depreciation causes it to shrink. 2. L is no longer fixed: population growth causes it to grow. 3. The consumption function is simpler.
- 15. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 15 How Solow model is different fromHow Solow model is different from Chapter 3’s modelChapter 3’s model 4. No G or T (only to simplify presentation; we can still do fiscal policy experiments) 5. Cosmetic differences.
- 16. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 16 The production functionThe production function In aggregate terms: Y = F (K, L ) Define: y = Y/L = output per worker k = K/L = capital per worker Assume constant returns to scale: zY = F (zK, zL ) for any z > 0 Pick z = 1/L. Then Y/L = F (K/L , 1) y = F (k, 1) y = f(k) where f(k) = F (k, 1)
- 17. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 17 The production functionThe production function Output per worker, y Capital per worker, k f(k) Note: this production function exhibits diminishing MPK. Note: this production function exhibits diminishing MPK. 1 MPK =f(k +1) – f(k)
- 18. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 18 The national income identityThe national income identity Y = C + I (remember, no G ) In “per worker” terms: y = c + i where c = C/L and i = I/L
- 19. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 19 The consumption functionThe consumption function s = the saving rate, the fraction of income that is saved (s is an exogenous parameter) Note:Note: ss is the only lowercase variableis the only lowercase variable that is not equal tothat is not equal to its uppercase version divided byits uppercase version divided by LL Consumption function: c = (1–s)y (per worker)
- 20. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 20 Saving and investmentSaving and investment saving (per worker) = y – c = y – (1–s)y = sy National income identity is y = c + i Rearrange to get: i = y – c = sy (investment = saving, like in chap. 3!) Using the results above, i = sy = sf(k)
- 21. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 21 Output, consumption, and investmentOutput, consumption, and investment Output per worker, y Capital per worker, k f(k) sf(k) k1 y1 i1 c1
- 22. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 22 DepreciationDepreciation Depreciation per worker, δk Capital per worker, k δk δ = the rate of depreciation = the fraction of the capital stock that wears out each period δ = the rate of depreciation = the fraction of the capital stock that wears out each period 1 δ
- 23. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 23 Capital accumulationCapital accumulation The basic idea:The basic idea: Investment makesInvestment makes the capital stock bigger,the capital stock bigger, depreciation makes it smaller.depreciation makes it smaller. The basic idea:The basic idea: Investment makesInvestment makes the capital stock bigger,the capital stock bigger, depreciation makes it smaller.depreciation makes it smaller.
- 24. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 24 Capital accumulationCapital accumulation Change in capital stock = investment – depreciation ∆k = i – δk Since i = sf(k) , this becomes: ∆k = s f(k) – δk
- 25. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 25 The equation of motion forThe equation of motion for kk the Solow model’s central equation Determines behavior of capital over time… …which, in turn, determines behavior of all of the other endogenous variables because they all depend on k. E.g., income per person: y = f(k) consump. per person: c = (1–s)f(k) ∆k = s f(k) – δk
- 26. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 26 The steady stateThe steady state If investment is just enough to cover depreciation [sf(k) = δk ], then capital per worker will remain constant: ∆k = 0. This constant value, denoted k* , is called the steady state capital stock. ∆k = s f(k) – δk
- 27. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 27 The steady stateThe steady state Investment and depreciation Capital per worker, k sf(k) δk k*
- 28. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 28 Moving toward the steady stateMoving toward the steady state Investment and depreciation Capital per worker, k sf(k) δk k* ∆k = sf(k) − δk depreciation ∆k k1 investment
- 29. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 30 Moving toward the steady stateMoving toward the steady state Investment and depreciation Capital per worker, k sf(k) δk k*k1 ∆k = sf(k) − δk ∆k k2
- 30. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 31 Moving toward the steady stateMoving toward the steady state Investment and depreciation Capital per worker, k sf(k) δk k* ∆k = sf(k) − δk k2 investment depreciation ∆k
- 31. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 33 Moving toward the steady stateMoving toward the steady state Investment and depreciation Capital per worker, k sf(k) δk k* ∆k = sf(k) − δk k2 ∆k k3
- 32. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 34 Moving toward the steady stateMoving toward the steady state Investment and depreciation Capital per worker, k sf(k) δk k* ∆k = sf(k) − δk k3 Summary: As long as k < k*, investment will exceed depreciation, and k will continue to grow toward k*. Summary: As long as k < k*, investment will exceed depreciation, and k will continue to grow toward k*.
- 33. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 35 Now you try:Now you try: Draw the Solow model diagram, labeling the steady state k* . On the horizontal axis, pick a value greater than k* for the economy’s initial capital stock. Label it k1. Show what happens to k over time. Does k move toward the steady state or away from it?
- 34. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 36 A numerical exampleA numerical example Production function (aggregate): = = × = 1/ 2 1/ 2 ( , )Y F K L K L K L = = ÷ 1/ 21/ 2 1/ 2 Y K L K L L L = = 1/ 2 ( )y f k k To derive the per-worker production function, divide through by L: Then substitute y = Y/L and k = K/L to get
- 35. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 37 A numerical example,A numerical example, cont.cont. Assume: s = 0.3 δ = 0.1 initial value of k = 4.0
- 36. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 38 Approaching the Steady State:Approaching the Steady State: A Numerical ExampleA Numerical Example Year k y c i k ∆k 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395 2.096 1.467 0.629 0.440 0.189 Year k y c i k ∆k 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395 2.096 1.467 0.629 0.440 0.189 Assumptions: ; 0.3;y k s δ= = =
- 37. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 39 Approaching the Steady State:Approaching the Steady State: A Numerical ExampleA Numerical Example Year k y c i k ∆k 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395 2.096 1.467 0.629 0.440 0.189 4 4.584 2.141 1.499 0.642 0.458 0.184 … 10 5.602 2.367 1.657 0.710 0.560 0.150 … 25 7.351 2.706 1.894 0.812 0.732 0.080 … 100 8.962 2.994 2.096 0.898 0.896 0.002 … 9.000 3.000 2.100 0.900 0.900 0.000 Year k y c i k ∆k 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395 2.096 1.467 0.629 0.440 0.189 4 4.584 2.141 1.499 0.642 0.458 0.184 … 10 5.602 2.367 1.657 0.710 0.560 0.150 … 25 7.351 2.706 1.894 0.812 0.732 0.080 … 100 8.962 2.994 2.096 0.898 0.896 0.002 … 9.000 3.000 2.100 0.900 0.900 0.000 Assumptions: ; 0.3;y k s δ= = =
- 38. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 40 Exercise: solve for the steady stateExercise: solve for the steady state Continue to assume s = 0.3, δ = 0.1, and y = k 1/2 Use the equation of motion ∆k = s f(k) − δk to solve for the steady-state values of k, y, and c.
- 39. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 41 Solution to exercise:Solution to exercise: 0 def. of steady statek∆ = and * * 3y k= = ( * ) * eq'n of motion with 0s f k k k= =δ ∆ 0.3 * 0.1 * using assumed valuesk k= * 3 * * k k k = = Solve to get: * 9k = Finally, * (1 ) * 0.7 3 2.1c s y= − = × =
- 40. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 42 An increase in the saving rateAn increase in the saving rate Investment and depreciation k δk s1 f(k) * k 1 An increase in the saving rate raises investment… …causing the capital stock to grow toward a new steady state: s2 f(k) * k 2
- 41. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 43 Prediction:Prediction: Higher s ⇒ higher k* . And since y = f(k) , higher k* ⇒ higher y* . Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run.
- 42. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 44 Egypt Chad Pakistan Indonesia Zimbabwe Kenya India Cameroon Uganda Mexico Ivory Coast Brazil Peru U.K. U.S. Canada France Israel GermanyDenmark Italy Singapore Japan Finland 100,000 10,000 1,000 100 Income per person in 1992 (logarithmic scale) 0 5 10 15 Investment as percentage of outpu (average 1960–1992) 20 25 30 35 40 International Evidence on InvestmentInternational Evidence on Investment Rates and Income per PersonRates and Income per Person
- 43. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 45 The Golden Rule: introductionThe Golden Rule: introduction Different values of s lead to different steady states. How do we know which is the “best” steady state? Economic well-being depends on consumption, so the “best” steady state has the highest possible value of consumption per person: c* = (1–s) f(k* ) An increase in s • leads to higher k* and y* , which may raise c* • reduces consumption’s share of income (1–s), which may lower c* So, how do we find the s and k* that maximize c* ?
- 44. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 46 The Golden Rule Capital StockThe Golden Rule Capital Stock the Golden Rule level of capital, the steady state value of k that maximizes consumption. * goldk = To find it, first express c* in terms of k* : c* = y* − i* = f (k* ) − i* = f (k* ) − δk* In general: i = ∆k + δk In the steady state: i* = δk* because ∆k = 0.
- 45. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 47 Then, graph f(k* ) and δk* , and look for the point where the gap between them is biggest. The Golden Rule Capital StockThe Golden Rule Capital Stock steady state output and depreciation steady-state capital per worker, k* f(k* ) δ k* * goldk * goldc * * gold goldi kδ= * * ( )gold goldy f k=
- 46. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 48 The Golden Rule Capital StockThe Golden Rule Capital Stock c* = f(k* ) − δk* is biggest where the slope of the production func. equals the slope of the depreciation line: steady-state capital per worker, k* f(k* ) δ k* * goldk * goldc MPK = δ
- 47. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 49 The transition to theThe transition to the Golden Rule Steady StateGolden Rule Steady State The economy does NOT have a tendency to move toward the Golden Rule steady state. Achieving the Golden Rule requires that policymakers adjust s. This adjustment leads to a new steady state with higher consumption. But what happens to consumption during the transition to the Golden Rule?
- 48. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 50 Starting with too much capitalStarting with too much capital then increasing c* requires a fall in s. In the transition to the Golden Rule, consumption is higher at all points in time. * * If goldk k> timet0 c i y
- 49. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 51 Starting with too little capitalStarting with too little capital then increasing c* requires an increase in s. Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption. * * If goldk k< timet0 c i y
- 50. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 52 Population GrowthPopulation Growth Assume that the population--and labor force-- grow at rate n. (n is exogenous) L n L ∆ = EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02). Then ∆L = n L = 0.02 × 1000 = 20, so L = 1020 in year 2.
- 51. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 53 Break-even investmentBreak-even investment (δ +n)k = break-even investment, the amount of investment necessary to keep k constant. Break-even investment includes: δ k to replace capital as it wears out n k to equip new workers with capital (otherwise, k would fall as the existing capital stock would be spread more thinly over a larger population of workers)
- 52. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 54 The equation of motion forThe equation of motion for kk With population growth, the equation of motion for k is ∆k = s f(k) − (δ + n) k break-even investment actual investment
- 53. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 55 The Solow Model diagramThe Solow Model diagram Investment, break-even investment Capital per worker, k sf(k) (δ +n )k k* ∆k =s f(k) − (δ +n)k
- 54. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 56 The impact of population growthThe impact of population growth Investment, break-even investment Capital per worker, k sf(k) (δ +n1)k k1 * (δ +n2)k k2 * An increase in n causes an increase in break- even investment, leading to a lower steady-state level of k.
- 55. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 57 Prediction:Prediction: Higher n ⇒ lower k* . And since y = f(k) , lower k* ⇒ lower y* . Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.
- 56. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 58 Chad Kenya Zimbabwe Cameroon Pakistan Uganda India Indonesia Israel Mexico Brazil Peru Egypt Singapore U.S. U.K. Canada FranceFinland Japan Denmark Ivory Coast Germany Italy 100,000 10,000 1,000 100 1 2 3 40 Income per person in 1992 (logarithmic scale) Population growth (percent per year (average 1960–1992) International Evidence on PopulationInternational Evidence on Population Growth and Income per PersonGrowth and Income per Person
- 57. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 59 The Golden Rule with Population GrowthThe Golden Rule with Population Growth To find the Golden Rule capital stock, we again express c* in terms of k* : c* = y* − i* = f (k* ) − (δ +n) k* c* is maximized when MPK = δ + n or equivalently, MPK − δ = n In the GoldenIn the Golden Rule Steady State,Rule Steady State, the marginal product ofthe marginal product of capital net ofcapital net of depreciation equals thedepreciation equals the population growth rate.population growth rate. In the GoldenIn the Golden Rule Steady State,Rule Steady State, the marginal product ofthe marginal product of capital net ofcapital net of depreciation equals thedepreciation equals the population growth rate.population growth rate.
- 58. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 60 Chapter SummaryChapter Summary 1. The Solow growth model shows that, in the long run, a country’s standard of living depends positively on its saving rate. negatively on its population growth rate. 2. An increase in the saving rate leads to higher output in the long run faster growth temporarily but not faster steady state growth.
- 59. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 61 Chapter SummaryChapter Summary 3. If the economy has more capital than the Golden Rule level, then reducing saving will increase consumption at all points in time, making all generations better off. If the economy has less capital than the Golden Rule level, then increasing saving will increase consumption for future generations, but reduce consumption for the present generation.
- 60. CHAPTER 7CHAPTER 7 Economic Growth IEconomic Growth I slide 62

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