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foundations macroeconomics

Introducing Advanced Macroeconomics: Growth And Business Cycles 1st Edition Peter Birch Sørensen, Hans Jørgen Whitta-Jacobsen - Solutions

2. Still using standard notation and definitions, e.g. fO.
1. Show that, at any time, the technological growth rate, gA"" A/A, according to this model is:gA = ~" 1 ~ = pA1'-\sRL)' .Explain for each of the two cases if> = 1 and < 1 how the output of the research sector A, the technological level A, and the technological growth rate g A evolve over time when
5. Analyse the resulting model as we have analysed models of semi-endogenous growth in this chapter going through the following steps: find the aggregate production function and describe its returns to scale. Find the transition equation for k,e k,/ A 1"" KJ (A ,L 1 ), verify convergence to a
4. What value of r maximizes the rate of economic growth {for the case A 1 = aG, as well as for A,= a{G,/L,))? Describe the opposing forces that determine the growth-maximizing tax rate.Discuss the policy implications of your analysis. How would policy implications be affected, do you think, if
3. Is there a scale effect? Assume alternatively that it is government expenditure per worker, G ,/L,, that is productive, so that A,=aG, above is replaced by A,= a{G,/L,). Analyse the model and find the growth rate of GOP per worker. Comment with respect to the scale effect and the aggregate
2. Show that the model is equivalent to an AK model (like (20) and (21) in the chapter) with appropriate relabelling of parameters. Find the growth rate of capital per worker and of GOP per worker that the model gives rise to.
1. Show that the aggregate production function (where it is taken into account how A, depends on Y, through taxation, etc.) is:Y1= (arL)faK,, and comment.
5. Show that the model can be condensed to the two equations:,= AK~ A "' L ( 1- a )/a1 K,. 1 = sY,+ (1- (5)K, .Find the growth rate, g •' of output per worker. Comment, for instance, with respect to scale effects.
4. Find the expression for the growth rate, g se• of output per worker in steady state. Comment with respect to what creates growth in output per worker in this model (when < 1 ).
3. Show that the transition equation for k1 is:k = __ k[s/(a - 1+(1 -0)](1-
2. Show that in this model:A (K a
1. Show that the aggregate production function in this model is:Why have we assumed < 1 /(1 -a)? Show that the aggregate production function has increasing returns to (K, L 1) whenever ¢> 0. Show also that if tJ> = 1 it has constant returns to K1 alone.
4. According to this model, growth in GOP per worker depends positively not only on sK as tested in Fig. 8.4, but also on sH. To test this create a diagram that plots growth rates in GOP per worker, gi, against investment rates in human capital, s~ across countries i with data taken from Table A
3. Show that according to the model above the growth rates of physical and human capital, respectively, in any period tare:K - K,. , -K, g, = K, H - H,. ,-H, g, = H, and then show that as x1= K,/H1 tends to its long-run equilibrium value x*, these two growth rates tend to the common value:g = s;.s
2. Define the ratio between physical and human capital, x1"" K, /Hr Show that the model above implies the following dynamic equation for this ratio:and demonstrate that this equation has properties that imply that in the long run x1 converges tox* =sKfsw
1. First show that if the production function of the individual firm is:Y, = (K'/)"(H'()'"(A.J_'/) '-"-"', and an external productive effect implies:then if = 1, the aggregate production function above results, where A= L 1- u - "' and v = a/(a + f!J). How should we view this model bearing in mind
5. By considering again the linear approximation around steady state, etc., find, also for this model, the rate of convergence A for y, according to the linear approximation. Show that(again) A goes to 0 as t/> goes to 1. (The latter result means that the endogenous growth model (¢; = 1) without
4. Show that the constant levels of capital per worker and output per worker in steady state are:* ( S )1/[(1- a)(l - 9\)J k = -n+O and* = (-s-)la+~(l a)J/ (1 ")(1 )J.y n+J
3. What is the aggregate production function in this model? Is labour unproductive at the aggregate level (also when¢ < 1 )? Why is it that in this case there is no positive, semi-endogenous type of growth in GOP per worker in steady state?The fact that the presence of the external effects does
2. Find the steady state values, k"' and f", for k, and Yu respectively. Show that in steady state the common growth rate of A" k, andy, is 0 (also when n> 0).
1. Show that now:(~\A,., _l..!il_ A, - (1 + n) ~ 'and that with the usual definitions, the transition equation is:k,. , = -- k,(sk;'- 1 - ( 1 )' - + (1 -b))~ . 1 + n Convergence to positive steady state values, k"" and j *, follows from the properties of the transition equation the same way as in
2. Show that the rate of convergence (for tJ> < 1) is:. (1 +n) W -91> -(1-o)A= (1- aj(1 - ¢) (1 + n) 9~>= (1 - a)(1-¢)[ 1-(1 11 ~ -J].Convince yourself that in the special case tJ> = 0, this rate is in accordance with the rate found in Chapter 5. Show that as¢> goes to 1, A goes to 0. Try to
1. Go through all the operations that lead to this equation yourself. (Of course, everything is the same as in Chapter 5 since there is no difference in the generic equation studied. It is important, though, that you can do the full linear approximation yourself.) Explain in what sense A is a
3. For each value of ¢, let the economy be initially in steady state for o = 0.08. After some periods, 1 0 say, o shifts permanently down too'= 0.06. Simulate the model over the periods before the shift and 200 periods after to create series (K1) and (Y1) . In one diagram plot the growth rate (Y1-
2. For each value of ¢>, compute the steady state values for K, and Y, before and after the shift in o.
1. Show that the steady state values, K* and Y*, of capital and output (also per worker), respectively, are:(s)l/{(1- a)(l - 91)1 K* =-c5 and What is the steady state value of A1?(s )la+91( 1- a)]/(( 1- a)( 1- 91))Y* =-0 .In the following let a = ~and s = 0.24 everywhere. Let o = 0.08 initially and
3. Assume that labour's share, {J + rp, is 0.6 with raw labour and human capital hav n~ equal shares fJ and rp, respectively, and that capital's share is 0.2, while land's and oil's shares are both 0.1 (as we assumed in this chapter). Rewrite the formula for gr using these values and comment with
2. Define the physical capital- output ratio, z, e K,!Y,, as well as the human, q 1 e H,!Y, . Show that along a balanced growth path where both z1 and q 1 are constant, the approximate growth rate of income per worker is:{3 K+E f.gr E;f g- n- sE. {3+K+E {J +K+ E {3+K+f(Hint: 1 -a- cp= f3 + K +
1. W ith usual notation like y 1 e Y, / L1, k1 e K,/L1, h1 e H1/ L1 , x1 e X/ L1, e 1e E1/ L11 and g{ e In y,- ln y,_,, etc., show that the per capita production function is:and that:g{ E;f ag~ + cpg~ + {3g - (1< +l:)n-ESE.
Exercise 9. Further numerical evaluation of g Y in the model with both land and oil Consider equation (35), which assumes a= 0.2, f3 = 0.6, K = r. = 0.1, and (36), which assumes further n = 0.01 and sE= 0.005. In the text we reminded you that a considerably higher population growth rate, e.g. n =
3. Find expressions for the approximate growth rates, g', g "' and gu, of the real interest rate, r11 the real wage rate, w11 and the real oil price, u11 respectively, in steady state. How does u 1 evolve compared to w1
2. Consider now only the steady state of the same model. Show that the approximate growth rates of Y1and K1 fulfil:Y K- fJ C g = g =--(g+n)- --SE.fJ +c fJ +t:Show that there is balanced growth in steady state (also) in the respects mentioned in Exercise 1, Question 2.
1. Consider the Solow model with oil. State expressions for the factor reward rates, r1 , w1 and u 11 as depending, in any period, on the state variables K1, L11 A 1 and R1• Show that the income shares are constant, in particular that the share of energy (or oil), u1E1/Y,= u1sER1/ Y1, is equal to
4. Show that the grow1h rate of y in steady state is exactly:{Jg -Kn f3 +K o and compare this to the approximate grow1h rate, gY, found in Section 1 of this chapter.
3. Show that the steady state value, z*, for the capital- output ratio is:z* = s(]3 ~ K) (n + g) + b 'and show that the above differential equation implies convergence of z to z*. (For the latter you can just draw t as a function of z or you can solve the differential equation as in Exercise 3 of
2. Show that the law of motion for z following from the model is the following linear differential equation in z:t = (/3 + K )s - /..z, where J...,. (f3 + K)(5 + f3(n +g). (Hint: Start from the expression for z you found in 1. Take logs and differentiate with respect to time. In the equation you
1. Show that the per capita production function and the capital-output ratio (still in obvious notation, e.g. x ""' X/L) are, respectively:y = k" AIIx', z = k 1- a A-11 x-•.
3. Assume that n > 0, but g = 0, more or less as considered by the classical economists.Describe how the real rates, r,, w, and v, evolve over time. If there is a fixed number of landlords owning the land, how will their life conditions evolve compared to the conditions of the workers?
2. Show that the steady state of the Solow model with land is in accordance with balanced growth, not only in the respect that the capital- output ratio is constant, but also in the following respects: the real wage rate, w,, grows by the same rate as both output per worker and capital per worker,
1. Show that in the Solow model with land the exact (in contrast to the approximate), common growth rate gr• of output per worker and of capital per worker in steady state is:gr• = (1 + g)~/(~H) -- - 1.(1 )K/(/1+•)1 + n(Hint: Start again from the per capita production function (8), this time
Exercise 4. Fighting structural unemployment Try to think of specific policies which might reduce the natural rate of unemployment. Explain why you think that these particular policies might reduce long-run unemployment.
Exercise 1 abo'le pointed to an{other) empirical shortcoming of the Solow model without human capital. The model's prediction that the investment rate in physical capital and the population growth rates should affect GOP per worker equally strongly (but in opposite directions) does not seem to hold
Exercise 10. An empirical shortcoming of the Solow model with human capital?
Exercise 9. An alternative Solow model with human capital You may have been wondering why we have built human capital into the model of this chapter in a way that is different from how it was brought into the picture in Chapter 5, Section 5 on growth accounting. In fact, there is an alternative
Exercise 7. Inferring a and qJ from the estimation of the convergence equation First find a way to estimate values fora. and cp from the information gathered in (39), (40) and(41 ). Then show that the implied values are a= 0.38 and cp= 0.32. Find the 95 per cent confidence intervals for the
Exercise 6. The effects of various parameter changes Let a base scenario be defined by the economy being, in all periods from 0 to 200 say, in steady state of the Solow model with human capital at parameter values c1. = (/) = . sK = 0.1 2, sH = 0.2, c5 = 0.055, n = 0, g= 0.02, and with an initial
Exercise 5. Further simulations to investigate stability In Section 3 we ran a simulation of the Solow model with human capital to check stability of k,. h, towards the steady state k*, h*. We only considered one starting point,k0 = 16, h0 = 2, which resulted in convergence to steady state, but in
Exercise 3. Balanced growth in steady state Show that growth in the steady state of the Solow model w ith human capital in a// respects accords w ith the concept of balanced growth. In particular, derive an expression for the steady state real rental rate, r*, and show that this is constant. Also
Exercise 2. Government in the Solow model with human capital When we were setting up the basic Solow model in Chapter 3 we were careful to explain that the savings rate, s, in S, = sY, could be interpreted as s = (1- cP){1 - c9), where we now assume if= 0, and constant values for the private
Exercise 1. One more empirical improvement obtained by adding human capital We opened this chapter by summarizing two main empirical shortcomings of the general Solow model. One was that its steady state prediction tends to understate the influences of saving and population growth rates on GOP per
Exercise 8. Growth accounting: have the growth miracle economies experienced above-normal TFP growth?Table 5.3 shows GOP per worker, y1, capital per worker, k, and average years of education in the population aged 15 or more, for some East Asian and some Western countries for 1961 and 1990.
Exercise 7. Growth accounting: does your country keep up?Table 5.2 shows GOP per worker, y, and non-residential capital per worker, k, for some countries and years. Use formula (45), assuming a = ~. to decompose for each country the average annual growth rate of GOP per worker over the full period
Exercise 6. Deriving an estimate of a from the estimation of the convergence equation In this chapter we derived an estimate of a from our estimation of the steady state regression equation (29). The y in that equation was estimated to be 1.47 (see the legend for Fig. 5.7), and since y = a/(1 -a)
Exercise 5. The effects of an increase in the rate of technological progress Assume that the economy is initially in steady state at parameter valuesa, s, n, g,b. Then from some period the rate of technological progress increases permanently to a new and higher constant level, g' > 0, but both n
4. Do all of the above once more, but now with g = 0.04 throughout. So, the base scenario is now given by a = ~. s = 0.12, n = 0.01, g = 0.04, J = 0.05 and A0 = 1, with the economy being in steady state at these parameter values in all periods, while the alternative scenario is given by an increase
3. From your simulations find the (approximate and rounded up) number of periods it takes forji 1 to move half the way from its old to its new steady state value. In Exercise 3 you were asked to derive a formula for the half-life, h, given a rate of convergence, A. Relate the half-life you have
2. Simulate the model using appropriate parameters and initial values, thus creating computed time series for relevant variables according to the alternative scenario. Extend Diagram 1 so that it shows the evolution in ji, both according to the base scenario and according to the alternative
1. Compute the s1eady state values k* and ji* for k1 and ji1 in the base scenario (all definitions are standard and as in the chapter). Illustrate the base scenario in three diagrams. Diagram 1 should plotji1 against timet (the points are just situated on a horizontal line). Diagram 2 should plot
3. Now derive the steady state value z'" directly from (51). Sketch the transition diaQram for z,, and establish properties of the transition equation that imply global, monotone convergence to z*, from any given initial z 0 > 0. (For this you must establish properties as listed in Footnote 8 in
2. Show, using the per capita production function, that z,= (k,/A,) ' - ".Then show, by writing this equation for period t + 1, using the definition k,. 1 "' K,. ,I L ,. ,. and then using the model's equations, that:From this equation show by appropriate manipulations that the law of motion for z,
1. Show that z, as just defined fulfils: z, = k,/y, =k,/y, = i
5. To solve the general Solow model analytically one can apply a trick as in Exercise 3 of Chapter 3. Define z "' k1 - a . Show that z is the capital- output ratio, k/y. Then show that:z = (1 -a)s-)z, {50)where A."' {1 - a)(n + g + o). Note that {50) is linear. Show that the steady state value of z
4. Show that {it follows from the model that) the growth rate ink at any time is:k - ..,.. = sk"- 1 -(n + g + o). k Illustrate this in a modified Solow diagram. Show that the growth rate, Yfy, of output per worker at any time t is a times the growth rate of k plus g. Assume that initially k0 <
3. Compute the steady state values, k* and Ji*, for capital per effective worker and income per effective worker. Compare to the parallel expressions found in this chapter for the model in discrete time. Also compute the steady state value and path {respectively), r* and w*, for the real rental
2. Illustrate the above Solow equation in a Solow diagram with k along the horizontal axis.Demonstrate {from the Solow diagram) that from any initial value, k0 > 0, capital per effective worker, k, will converge towards a specific steady state level, k*, in the long run (as t ->=).
1. Applying methods similar to those used in Exercise 2 of Chapter 3, show that the law of motion fork following from the general Solow model in continuous time can be expressed in the Solow equation:k = gj(a - (n + g + o)k Compare to {19) for the model in d iscrete time.
6. Show that the steady state values for GOP per capita, the wage rate, wealth per capita and national income per capita all decrease in response to the increase in 1:. (Hint: You don't have to solve the model for y"*. From the shift of the transition curve you can infer how steady state wealth per
5. Show that an increase in 1: implies a decrease in w* + rk*, whenever 1: > 0 initially. How is the transition equation affected by an increase in r ?
4. Using the above expressions fo r k* and w*, show that:w* +rk* = 1 +-_---(1 -a) -- - . [1: a ] ( a )a/(1 - a)r+r 1 -a r+r
3. Show that the transition equation for national wealth per capita is:1 + sr s(w* + r k*) v = --v + --'-----'- /+ 1 1 + n 1 1 + n
2. Show that domestic capital per capita and the wage rate, respectively, adjust immediately to:* ( a ')1/(1-a) k =-f'+E'and w* = (1 -a) -_- . (a )'l/(1 - a)f+ E
1. W rite down the complete model (for your convenience) and show that in any period t, GOP per capita is:y, = w1+(l'+ E)k1,
Exercise 7. Long-run national income in the small open economy as depending on domestic and international savings propensities A certain restatement of (27) can be of interest. As argued in this chapter, the world economy is a closed system, so it should behave according to the model of Chapter 3
4. Now simulate each of the transition equations from period zero and onwards (for a number of periods of your own choice, but be sure to have quite a few), thereby creating a sequence (v 1)for the open economy and a sequence (k1) for the closed. From these sequences compute the associated
3. Compute the steady state values, y n* = r: and u* = k;, under these specifications. Assume that both economies, the open and the closed, start in their steady states and then, as from period one, say, the total factor productivity increases permanently up to 8' = 2. Compute the new steady state
2. Consider the transition equation for the open economy (v,. , as a function of v,) and show that under our assumptions its slope is:1 + (J.n 1 + n
1. Show from (27) that under these assumptions steady state national income per capita is:Then show, by comparing to the relevant formula in Chapter 3, that this y"* is equal to the steady state GOP per capita, y~, for the closed economy. Show further that steady state national wealth per capita,
4. How is the long-run functional income distribution in the open economy affected by a permanent international interest rate shock, where r increases permanently to a new and higher level? Explain your result.
1. Show, using (16) and the fact that w, jumps immediately tow*, that labour's share in any period tis:w, u, v, - = 1-r- = 1-r--- Y7 Y~ w* + rv, 'and show that as national wealth, v,, converges towards its steady state level, v*, from below(above), labour's share will converge to its steady state
Exercise 3. The income distribution in the small open economy As stated in this chapter, in any period t, the share of labour income in GOP, w,/y, is equal to V'1 -a., while the share of income earned on domestic capital in GOP, rk,/y,, is a.. These features are unchanged from the model for the
Exercise 2. Golden rule in the open economy?Show that in the small open economy's steady state consumption per capita, is:1 c* = (1 -s) w*1 - (s/n)r 'where the expression for w* of ( 19) could be inserted (but, since w* does not depend on s, this is not of importance for the present exercise). Show
Exercise 1. The elasticity of long-run national income per capita with respect to the savings rate in the small open economy This exercise investigates how strongly national income per capita is influenced by the savings rate in the open economy compared to the closed. First show from (27) that the
Exercise 11. Endogenous population growth and club convergence in the basic Solow model Consider a growth model that consists of the same equations as the basic Solow model except that then in Eq. (19) is no longer exogenous, but rather an endogenous variable that depends on prosperity. Hence, in
Exercise 1 0. The recovery of Japan after the Second World War Table 3.2 shows GDP per worker for the USA and Japan for the period 1959 to 1996, measured in a common, purchasing power-adjusted currency (1996 dollars). The table only goes to 1996 to avo d considering (in the present long-run
Exercise 9. Further empirically testing the steady state prediction of the basic Solow model This exercise anticipates some issues taken up in succeeding chapters, and it is a very useful empirical exerc se. The data you will need can be taken from Table A. In Figs 3. 7 and 3.8 we tested the steady
Exercise 8. Balanced growth in the steady state Show that all the requirements for balanced growth, as these are stated in Chapter 2, are fulfilled in the steady state of the basic Solow model. Why are we, nevertheless, not completely satisfied with the basic Solow model's steady state as a
Compute the particular value k** of k, that maximizes this distance. Illustrate the point in your Solow diagram: the slope of Bk;' should equal the slope of (n + o)k,. Explain this.What is the requirement on s to ensure that the steady state capital intensity, k*, becomes equal to the golden rule
Exercise 7. Golden rule In this chapter it was stated that the golden rule savings rate that maximizes steady state consumption per person as given by (35) is s** = a . Show this. To illustrate, draw a Solow diagram with the curve, sBk , and the ray, (n + c5)k1, as usual, and draw the curve Bk;' as
Exercise 6. A one shot increase in L1 Analyse in the Solow diagram, the qualitative effects of a one shot increase in L,, starting in steady state at the prevailing parameter values. The event we are interested in here is that at the beginning of some period, period one say, L, increases once and
Exercise 5. The effects of an increase in technology Do all you were asked to do in the previous exercise (there for a decrease in n) only this time for an increase in B. First analyse the qualitative effects on the Solow diagram, explaining the transition mechanisms. Then consider the numerical
4. In Chapter 2 it was shown that developed economies have experienced relatively constant positive growth rates in GOP per worker (annual rates around 2 per cent) and relatively constant capital- output ratios in the long run. Show that developed countries must then also have experienced
3. Show that with the CES production function, the income share of labour under competitive market clearing is:wL 1 -a - = ---,-,----y (K)(
2. Find the marginal rate of substitution between capital and labour for the CES production function, and show that the elasticity of substitution is -a, independent of rand w.The CES function is more general than the Cobb- Douglas function, since the particular value a = 1 (or the limit for a ->
1. Find the marginal rate of substitution between capital and labour (the marginal product of capital divided by the marginal product of labour) for the Cobb- Douglas production function.Show that profit maximization given rand w requires that this marginal rate of substitution is equal to r/w, and
Exercise 7. Galton's Fallacy 14 It is an implication of convergence that a lower initial GOP per worker should, other things being equal, imply a higher subsequent growth in GOP pe· worker. This was used in the chapter to test convergence: the lack of a negative relationship between initial GOP
Exercise 6. Growth and trade This chapter has presented the stylized facts of growth that are most important for the theoretical developments in subsequent chapters. There is at leas! one more important fact, which relates growth to growth in trade. To draw the following figure you should go to the
Exercise 5. Level convergence among the rich?Commenting on Fig. 2.3 in Section 3 we said that it showed a tendency towards convergence to a common growth path with constant growth. Perhaps we were too hasty. The following figure will make level differences more visible if they are there. For the
Exercise 4. Growth against ultimo level In Section 3 there were figures showing across countries the log of GDP per worker in an initial year along the horizontal axis, and average annual growth in GDP per worker in a subsequent relatively long period along the vertical axis. For the OECD countries
Exercise 3. Time to double Assume that in a specific country GOP per capita, or per worker, grows at the rate g each year over many years, where g is written as a percentage, e.g. 2 per cent. Show that GOP per capita, or worker, will then approximately double every 70/g years. Use this to set up a
Exercise 2. The income distribution for the rich part of the world Table 2.4 shows GOP per worker {in 1996 US dollars) and population size for 1960 and 2000 for the rich part of the world (defined here as the OECD countries for which data were available).Construct Lorenz curves for the rich world

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