Introduction to Economic Growth Fall 2016
Introduction to Economic Growth Fall 2016
Introduction to Economic Growth Fall 2016 Introduction to Economic GrowthQuestionsName:QuestionsExam Version: 4331W QBQuestions for all chaptersâ¢ The point value for each question is the approximate number of minutes you should take to solve the question.This document has 16 pages (not including this front page), numbered 116. Good luck!Exam RecordCh.1:/ 3 pointsCh.2:/ 78 pointsCh.3:/ 150 pointsCh.4:/ 29 pointsCh.5:/ 91 pointsEconometrics:/ 63 pointsMRW:/ 9 pointsCh.6:/ 10 pointsCh.7:/ 20 pointsCh.8:/ 105 pointsCh.9:/ 0 pointsCh.10:/ 0 pointsTotal:/ 558 pointsIntroduction to Economic GrowthFall 2016QuestionsName:Chapter I. (3 points) Answer all questions in this part.(3pts )1. List three of the facts we discussed that we are attempting to explain with the Solow Model. (Note: Kaldorâsfacts will be one fact, even if you list multiple of them)Chapter II. (78 points) Answer all questions in this part.(5pts )1. Consider a onetime permanent increase (alternative: decrease) in the stock of capital. Describe the shortrunand longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You should discuss theeﬀect on growth rates and levels of capital and income. You may assume g = n > 0, i.e. we start in a steadystate.(5pts )2. Consider a permanent increase (alternative: decrease) in the savings rate. Describe the shortrun and longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You should discuss the eﬀect ongrowth rates and levels of capital and income. You may assume g = n > 0, i.e. we start in a steady state.(5pts )3. Consider a onetime permanent increase (alternative: decrease) in the stock of technology. Describe theshortrun and longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You shoulddiscuss the eﬀect on growth rates and levels of capital and income. You may assume g = n > 0, i.e. we startin a steady state.(5pts )4. Consider a onetime permanent increase (alternative: decrease) in the growth rate of technology. Describethe shortrun and longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You shoulddiscuss the eﬀect on growth rates and levels of capital and income. You may assume g = n > 0, i.e. we startin a steady state.(5pts )5. Consider a onetime permanent increase (alternative: decrease) in the stock of labor. Describe the shortrunand longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You should discuss theeﬀect on growth rates and levels of capital and income. You may assume g = n > 0, i.e. we start in a steadystate.(5pts )6. Consider a onetime permanent increase (alternative: decrease) in the growth rate of labor. Describe theshortrun and longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You shoulddiscuss the eﬀect on growth rates and levels of capital and income. You may assume g = n > 0, i.e. we startin a steady state.E1/4331W – Question Bank(5pts )Name:7. Chapter 2, question 3(8pts )â Page 2 of 16 â8. Chapter 2, question 5(35pts )9. Solow Growth and Economic FactsThe following questions relate the Solow Growth model to the economic facts discussed in Chapter 1. Assumethe basic Solow Model given byâ¢ Firms use capital, K, and labor, L, to produce output, Y, according to the following production function.Y = K α L1−α , 0 ≤ α ≤ 1The economy starts oﬀ at t = 0, with K(0) = K0 and L(0) = L0 The labor force, which is equal tothe total population, grows at a constant rate, n.˙L=nLâ¢ Consumers save at a constant rate, 0 ≤ s ≤ 1, and consume the rest.â¢ Capital depreciates at a constant rate, 0 ≤ δ ≤ 1, giving rise to the following capital accumulationequation.˙K = sY − δKYou may assume we have proven a steady state of capital and output (k ∗ , y ∗ ) exists, and is given bysn+g+δ1/1−αk∗ =sn+g+δα/1−αy∗ =Note: Only the explanation is worth points for the following questions. A yes/no answer will be given a zero.(a) (5 pts) Does this model explain Fact 1, that there exists enormous variation in percapita income acrosscountries? Explain (using graphs and equations where appropriate).(b) (5 pts) Does this model explain Fact 2, that rates of economic growth vary across countries? Explain(using graphs and equations where appropriate).(c) (5 pts) Does this model explain Fact 3, that growth rates are not generally constant across time? Explain(using graphs and equations where appropriate).(d) (5 pts) Does this model explain Fact 51, that the real return to capital shows no trend? Explain (usingequations where appropriate).(e) (5 pts) Does this model explain Fact 52, that the share of income devoted to capital and labor show notrend? Explain (using equations where appropriate).E1/4331W – Question Bankâ Page 3 of 16 âName:(f) (5 pts) Does this model explain Fact 53, that the growth rate of output per person has been positiveand relatively constant over time? Explain (using graphs and equations where appropriate).(g) (5 pts) If your answer to the previous question was âno,â what simple modiﬁcation to the above modelwould allow it to match this fact? Explain. (You may assume the steady state of the new model exists,but please provide equations for them in your explanation).Chapter III. (150 points) Answer all questions in this part.(15pts )1. Chapter 3, Question 2(15pts )2. Chapter 3, Question 6(3pts )3. Deﬁne convergence. Does the data support absolute convergence of countries? Explain your answer.(15pts )4. Write the Solow model with skilled labor. Deﬁne the state variables, and solve for the steady state. Whichof the facts discussed in chapter 1 does this model match?(5pts )5. Consider the previous question. Describe three policies you would recommend to a developing country lookingto improve percapita income.(2pts )6. Consider the previous question. What policies can the developing country follow to permanently increaselongrun growth rates.(70pts )7. Solow GrowthThis problem requires you to work with the basic Solow Model. In this economy, there is a representativeconsumer and a representative ﬁrm.â¢ Firms use capital, K, and labor, L, to produce output, Y, according to the following production function.Y = K α L1−α , 0 ≤ α ≤ 1The economy starts oﬀ at t = 0, with K(0) = K0 and L(0) = L0 The labor force, which is equal tothe total population, grows at a constant rate, n.˙L=nLâ¢ Consumers save at a constant rate, 0 ≤ s ≤ 1, and consume the rest.E1/4331W – Question Bankâ Page 4 of 16 âName:â¢ Capital depreciates at a constant rate, 0 ≤ δ ≤ 1, giving rise to the following capital accumulationequation.˙K = sY − δK(a) (10 pts) Show that in the steady state, the level of per capita capital can be expressed askss =sn+δ11−α(b) (10 pts) Suppose the country in the Solow Model goes through a demographic transition and the population growth rate decreases from n to n . Does the steady state level of income per capita, decrease,increase, or stay the same? Justify your answer using an expression for yss as a function of the exogenousvariables and parameters of the model. Also, show how the steady state changes by using the Solowdiagram.(c) (10 pts) Draw a graph of the time series of per capita income before and after the change in the populationgrowth rate, paying careful attention to the transition dynamics.(d) (10 pts) Consider an economy where a portion of the population does not participate in the labor force.That is, there is some constant rate of unemployment, 0 < Âµ < 1. Denote the total population, P , sothatL(t) = (1 − Âµ)P (t)and the population grows at a constant rate,˙P= n, P (0) = P0PDerive an expression for yss as a function of s, δ, n, Âµ, and α.(e) (10 pts) Suppose the unemployment rate rises to Âµ > Âµ. Illustrate this change in the Solow diagram.What happens to the steady state level of per capita income?(f) (5 pts) What is a balanced growth path?(g) (5 pts) What is a state variable?(h) (10 pts) What was the primary motivation for adding technological progress to the Solow Growth Model?(25pts )8. Solow Growth:Consider the standard Solow growth model with aggregate output given byY = K α L1−αwhere K denotes aggregate capital, and L denotes labor. Capital depreciates at the rate d and consumerssave a fraction s of their total income.Capital evolves according to˙K = A1−α sY − dKE1/4331W – Question Bankâ Page 5 of 16 âName:where A denotes the technology of transforming investment into capital. Population and technology grow atconstant rates, given by˙˙AL= n, and=gLA. Denote capital and output per capita byk≡KY, and y ≡LLDenote capital and output per eﬃcient unit of labor byKYËk≡, and y ≡ËALAL(a) (5 pts) Use the equations and deﬁnitions above to obtainy = kα(b) (10 pts) Use the equations and deﬁnitions above to obtain˙ËkË= sk α−1 − (d + g + n)ËkË(c) (5 pts) Calculate the steady state values of k and yË(d) (5 pts) One of Kaldorâs facts is that, in the long run, K/Y is constant. Does this model deliver this fact?(Hint: Use the solution to the previous part to ﬁnd K ∗ and Y ∗ )Chapter IV. (29 points) Answer all questions in this part.(1pt )1. Deﬁne nonrivalrous goods.(1pt )2. Deﬁne excludability.(3pts )3. Using the ideas of rivalrous and nonrivalrous goods, explain why ideas have increasing returns to scale, butmost other goods have decreasing returns to scale.E1/4331W – Question Bank(3pts )5. Chapter 4, Question 2(15pts )Name:4. Chapter 4, Question 1(6pts )â Page 6 of 16 â6. Chapter 4, Question 3Chapter V. (91 points) Answer all questions in this part.(23pts )1. The Romer Model:You may assume the following equations:˙K = sk Y − δK˙L=nLÂ¯˙A = θLλAÂ¯θ = θAφLy + LA = L(a) (2 pts)What problem with earlier models does the Romer formulation address?(b) (2 pts)In chapter 4, we discussed how ideas have increasing returns to scale. Write down the productionfunction for the Romer model, and explain whether or not it exhibits increasing returns to scaleÂ¯(c) (2 pts)Explain what θ represents. Provide intuition on what φ is.(d) (2 pts)In the equation for technology change, provide intuition on what λ is.(e) (2 pts)Show that gy = gk = gA where lowercase letters are percapita variables.(f) (5 pts)Solve for the growth rate of A.˙(g) (5 pts)Consider the case where λ = 1 and φ = 0. What is A? Compare the Romer model to the typical˙˙LLSolow model under two conditions: 1) L = 0, and 2) L = n > 0(h) (3 pts)What eﬀect do government policies have on long run growth rates and levels?E1/4331W – Question Bankâ Page 7 of 16 âName:(5pts )2. Consider a onetime permanent increase (alternative: decrease) in the stock of capital. Describe the shortrunand longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You should discuss theeﬀect on growth rates, technology levels, and levels of capital and income. You may assume g = n > 0, λ =1, φ = 0.(5pts )3. Consider a permanent increase (alternative: decrease) in the savings rate. Describe the shortrun and longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You should discuss the eﬀect ongrowth rates, technology levels, and levels of capital and income. You may assume g = n > 0, λ = 1, φ = 0.(5pts )4. Consider a onetime permanent increase (alternative: decrease) in the stock of technology. Describe theshortrun and longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You shoulddiscuss the eﬀect on growth rates, technology levels, and levels of capital and income. You may assumeg = n > 0, λ = 1, φ = 0.(5pts )5. Consider a onetime permanent increase (alternative: decrease) in the stock of labor. Describe the shortrunand longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You should discuss theeﬀect on growth rates, technology levels, and levels of capital and income. You may assume g = n > 0, λ =1, φ = 0.(5pts )6. Consider a onetime permanent increase (alternative: decrease) in the growth rate of labor. Describe theshortrun and longrun eﬀects on the economy, using ﬁgures and equations where appropriate. You shoulddiscuss the eﬀect on growth rates, technology levels, and levels of capital and income. You may assumeg = n > 0, λ = 1, φ = 0.(5pts )7. Consider a onetime permanent increase (alternative: decrease) population share engaged in research. Describe the shortrun and longrun eﬀects on the economy, using ﬁgures and equations where appropriate.You should discuss the eﬀect on growth rates, technology levels, and levels of capital and income. You mayassume g = n > 0, λ = 1, φ = 0.(13pts )8. Romer vs SchumpeterIn this question, we want to compare the Schumpeter Model with the Romer model(a) (5 pts)Explain creative destruction, and the diﬀerence in how intermediate goods are treated in the Romermodel versus the Schumpeter model(b) (5 pts)Explain the process of innovation in Schumpeter. How does this diﬀer from the process of innovation in Romer?(c) (3 pts)What is the expected growth rate of technology over time? How does the actual growth rate diﬀerfrom that in Romer?E1/4331W – Question Bankâ Page 8 of 16 âName:(10pts )9. In both the Romer and Schumpeter models, R&D is not at an optimal level. Explain, in detail, two reasonswhy we donât achieve optimal results.(5pts )10. If the intermediategoodﬁrms in the Romer model did not have monopoly power, what would be the equilibrium price of a patent? Explain.(5pts )11. If the intermediategoodﬁrms in the Romer model did not have monopoly power, what would be the growthrate of output per capita in Romer model? Explain.(5pts )12. Provide an explanation, consistent with what you learned from the Romer model, for what happened duringthe Industrial Revolution that generated the virtually constant growth rates of output per capita the worldhas experienced since then.Econometrics (63 points) Answer all questions in this part.(1pt )Ë1. Give the equation that calculates the bias of the estimator, θ(1pt )2. Deﬁne a consistent estimator(1pt )3. Letâs say we run a regression and obtain the following: Total Sum Squares = 100, Explained Sum Squares =90, Sum Squared Residuals = 10. Find R2 . (Note: You must show work to receive credit)(40pts )4. Testing Models:Suppose we have data on the income of recent college graduates and the average income of their parents,and we would like to know if there is a relationship between the two. That is, we are asking whether there isa relationship between the starting salary of a recent college graduate and the income of his/her parents.The data set is as follows,â¢ n observationsâ¢ Yi , (Fatherâs Income + Motherâs Income)/2â¢ Xi , Annual Income of graduateâs ﬁrst jobTo determine if a relationship exists, we run the following regression,ËËYi = β0 + β1 Xi + uiin order to test the following hypothesis,H0 : β1 = 0, H1 : β1 = 0E1/4331W – Question Bankâ Page 9 of 16 âName:ËLet, Yi be the value predicted by the OLS regression line,ËËËYi = β0 + β1 Xi(a) (10 pts) Set up the minimization problem used to derive the O.L.S. estimators of β0 and β1 . (Just setup the problem. You do not need to derive the formulas)Ë(b) (10 pts) You run the regression in stata and obtain the following estimate of β1 and SE(β1 ),ËËβ1 = 1.02, and SE(β1 ) = 0.5ËInterpret the meaning of β1 .(c) (10 pts) Can H0 be rejected at a signiﬁcance level of 5%? How about at a signiﬁcance level of 1%?Brieﬂy explain how you came to your conclusion.Ë(d) (10 pts) What is omitted variable bias? Think of one omitted variable that could cause β1 to be biasedand explain why.E1/4331W – Question Bank(10pts )â Page 10 of 16 âName:5. Testing Models:Assume we have the following balanced growth path equationy ∗ (t) =skn+g+δα/1−αhA (t)where H is highskilled labor, generated from lowskilled labor, L.H = eφu L, h =Assume u = 10Hd log H, and=φLduE1/4331W – Question Bankâ Page 11 of 16 âName:(a) (4 pts) Write down the regression equation you would use to estimate the parameters α and φ.(b) (2 pts) Assume we are interested in the eﬀect of φ. What is our null hypothesis, H0 ? What is ouralternative hypothesis, H1 ? (Note: You must specify these with respect to this question.)Ë(c) (2 pts) Assume we obtain an estimate of φ, β = −0.2 with a standard error of 0.25. Do we accept orreject the null hypothesis? Why?(d) (2 pts) Consider the model with skilled labor studied in Chapter 3. What condition would we expect onour estimate of φ?(10pts )6. Evaluation of the Solow Model using OLSfSuppose you have the following data for each country i: output per capita in 2010, yi , output per capita in01950, yi , average growth rate of yi from 1950 to 2010, gi , average savings/investment rate from 1950 to2010, si , and average population growth rate from 1950 to 2010, ni . Also suppose you have the depreciationrate d and the growth rate of technology g.Consider the Solow Model with technology, given byY = K α (AL)1−α,˙K = sY − dK.˙L=nL(a) (5 pts) Consider the following regression:flog yiand˙A= g.A(1)(2)(3)= a + b1 log (si ) + b2 log (ni + d + g) + e1 ,iwhere e1 denotes the residuals. If the capital income share is equal to 1/3 what are the exact predictionsiof the Solow model (the simple version, described by equations (1), (2), and (3)) for the values of b1 andb2 ? You must use the above equations to derive your result.(b) (5 pts) Consider the following two regressions:0gi = a + c1 log yi + e3 ,i0and gi = a + d1 log yi + d2 log (si ) + d3 log (ni + δ + g) + e2 ,iwhere e2 and e3 denote residuals. If you run these regressions for all countries in the world what wouldiiyou ﬁnd about c1 and d1 ? Explain.E1/4331W – Question Bankâ Page 12 of 16 âName:MankiwRomerWeil (9 points) Answer all questions in this part.(9pts )1. MankiwRomerWeil:The following questions refer to the MankiwRomerWeil paper we discussed in class.(a) (3 pts) Table I on the last page shows the regression results obtained from the basic Solow model. In twoto three sentences, explain a problem the authors found in these results that motivated their decision toaugment the model. (Note: You must refer to the table in describing the problem)(b) (3 pts) Table II shows the regression results obtained from the augmented model. In two to threesentences, explain whether the problem discussed in the previous section is resolved by the augmentedmodel, using results from the table to support your answer.(c) (3 pts) Now consider Table V, which shows the results for testing for conditional convergence. For whichsets of countries do we see conditional convergence occurring? Support your answer using data from thetable.E1/4331W – Question Bankâ Page 13 of 16 âName:E1/4331W – Question Bankâ Page 14 of 16 âName:Chapter VI. (10 points) Answer all questions in this part.(10pts )1. Chapter 6, Question 5Chapter VII. (20 points) Answer all questions in this part.(5pts )1. Chapter 7, Question 1(10pts )2. Chapter 7, Question 3(5pts )3. Chapter 7, Question 4Chapter VIII. (105 points) Answer all questions in this part.(15pts )1. Poverty Traps Consider the following version of the Solow model, where output is given byY = K α L1−α .(B1)Suppose that consumption per capital, c, is given byc=y, if y < cc + (1 − s) (y − c) , if y ≥ c(B2)that is, consumers consume all income up to the subsistence level, c, and start to save a constant proportion,s, only for y ≥ c. Suppose that capital, K, evolves according to˙K = I − dK,(B3)where d is the capital depreciation rate and I denotes investment. Population, L, grows at a constant rateof n,˙L= n.(B4)LLet C denote aggregate consumption and denote per capita variables byc≡C,Lk≡K,Landy≡Y.L(B5)Further, notice that it follows from (B1) and (B5) thaty = kα .(B6)E1/4331W – Question Bankâ Page 15 of 16 âName:(a) (5 pts) Using equations (B2) and (B5), and the fact that I = Y − C, calculate I as a function ofparameters, Y and L, in both cases: if y < c and if y ≥ c.(b) (5 pts) Using equations (B3), (B5), and (B6), and the results from part B.1, obtain1− (d + n) ks (k α − c) − (d + n) k˙k=, if k < c α1, if k ≥ c α(B7)1(c) (5 pts) Using equation (B7), draw a Solow diagram and consider an economy that starts with k < c α .What would happen to this economy in the longrun? What would happen to this economy in the longrun1if it received an inﬂux of capital that made k jump to a level above c α ?(20pts )2. Chapter 8, Question 1(20pts )3. Chapter 8, Question 3(20pts )4. Chapter 8, Question 4(30pts )5. Solow Model and LandConsider the standard Solow growth model with aggregate output given byY = K α (AL)1−α,(1)where K denotes aggregate capital, L labor (or population) and A is the technology. Capital depreciates atthe rate d and consumers save a fraction s of their total income. The economy is closed, so savings equalsinvestment and capital evolves according to˙K = sY − dK.(2)Population and technology grow at constant rates n and g respectively, that is˙L=nLand˙A= g.AK,Landy≡Y.L(4)y≡ËY.AL(5)(3)Denote capital and output per capita byk≡Denote capital and output per eﬃcient units of labor byKËk≡,ALandAlso, let gx denote the balanced growth path growth rate of x, i.e. gx = x/x, where x can be any variable˙in the model.E1/4331W – Question Bankâ Page 16 of 16 âName:(a) (10 pts) Using the equations listed above, the appropriate derivations and the Solow diagram, show thatin the longrun the growth rate of K will be constant.(b) (5 pts) Using the fact that gK is constant and equation (2) show that K/Y will also be constant.(c) (5 pts) Use equation (1) to calculate gy , the balanced growth path growth rate of output per capita as afunction of parameters. [Hint: divide both sides of equation (1) by Y α and use the result from part 2]Now, suppose that land, T , is also a factor of production and that the production function isY = T γ K α A1−α L1−γ−α .(1 )(d) (5 pts) Use equation (1 ) to calculate the balanced growth path growth rate of output per capita, gy , asa function of parameters.(e) (5 pts) Your result in part 4 should have an additional term relative to the result in part 3. Explain why.Chapter IX. (0 points) No Questions at this timeChapter X. (0 points) Answer all questions in this part.

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