Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 3

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1 Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 3. (from Romer "Advanced Macroeconomics" Chapter ) Consider an economy with technological progress but without popultion growth that is on its balanced growth path. Now suppose there is a one-time jump in the number of workers. (a) At the time of the jump, does output per unit of e ective labor rise, fall, or stay the same? Why? At the time of the jump output per unit of e ective labor instantly falls, as well as capital per unit of e ective labor does. While the capital stock is unable to change instantly and technological progress does not alter other than before ~k = K AL must decrease. The same argument applies for ~y = AL. Intuitively, it is less capital per e ective worker available and output has to be shared by more "e ective workers". ~ = K AL 0 AL 0 (b) After the initial change (if any) in output per e ective labor when the new workers appear, is there any further change in output per unit of e ective labor? If so, does it rise or fall? Why? The one-time jump of L may be intepreted as a new initial situation for the economy that does not matter in the long run. The decrease in k ~ leads to a situation where actual investment exceeds Break-Even-investment. Intuitively a decrease in k ~ increases MP K and thus leads to MP K R. Firms therefore need to pay less for their capital than the capital yields for them which makes investment lucrative. Hence, the capital stock and thereby k ~ and ~y increase until the BGP-value of k ~ that was valid before the increase in L is obtained again. k ~ and ~y end up at the initial situation again. (c) Once the economy has again reached a balanced growth path, is output per unit of e ective labor higher, lower, or the same as it was before the new workers appeared? Why?

2 Due to the argumentation in (b) the economy ends up in its initial situation. This result refers to the fact that a jump in L does not alter any growth rate, savings rate or production function but simply e ects an initial condition. As one of the basic results of the Solow-Swan Analysis is that initial conditions of K, A and L do not matter, it is obvious that the economy moves to it s balanced growth path again. A comparable situation might be the destruction of capital (e.g. in a war) because this would also alter ~ k without a ecting growth rates of K, A and L. Figure. (from Romer "Advanced Macroeconomics" Chapter ) Consider a Solow economy that is on its balanced growth path. Assume for simplicity that there is no technological progress. Now suppose that the rate of population growth falls. (a) What happens to the balanced growth path values of capital per worker, output per worker and consumption per worker? Sketch the paths of this variables as the economy moves to its new balanced growth path. Figure BGP value of k : s f (k) = (n + ) k ) s f = k = k sf 0 (k) (n+) = k s(kf 0 (k) f(k)) 0 BGP value of @n @n 0 BGP value of c : c BGP = f (k) s f (k) = ( s) f = ( s) f 0 This gives us: Figure 3

3 (b) Describe the e ect of the fall in population growth on the path of output (that is, total output, not output per worker). We know that in the steady state, i.e. if the economy is on its BGP, in an economy where there is no technological progress k = K L and therefore y = L are constant. That means, that in this case total output has to grow at the same rate as the equilibrium population. Hence, total output grows at the growth rate n in the initial equilibrium and at the - decreased - growth rate n in the new steady state. Furthermore, we know that y = L increases if population growth decreases (part (a)). This results from the fact that the population instantly grows slower as the growth rate of the population drops. However, adjustment of the growth rate of total output is gradually as compared to the one time drop of the growth rate of the population, because as y = L increases total output has to grow at a rate n n n during the adjustment process. 3. Assuming that capital does not move internationally (as in the Solow model), international di erences in savings rate translate into high returns to capital in low savings economies. The Solow model assumes that economies are closed to international nancial markets. In reality, however, there would be a tendency for countries with high marginal products of capital (and therefore high real interest rates) to attract the savings of countries with low marginal product of capital (and therefore low real interest rates). To see how strong this tendency may be, assume that there are two countries that function as described in the Solow model. Assume that these countries are identical in everything, except their savings rates. In particular, country H has a savings rate that is twice that of country L. How di erent will the marginal products of capital and therefore the real interest rates of the two countries be once they have reached their balanced growth paths? Assume that at the initial position: L H = L L = L; A H = A L = A; n H = n L = n; H = L = ; a H = a L = a and s H = s L Assume also, for simplicity, that technology in both countries is described by identical Cobb-Douglas production functions. Then it is true that for each country, in the BGP (see Question ): k K AL = s n++a The real interest rate is the marginal product of capital minus depreciation: 3

4 r = MP K = (k). Then in the BGP, r = s n++a s n++a = Comparing the returns in both countries, we see that r L r h = sl n++a sl n++a = sl n++a > 0 This is the expected result that MPK will be higher in the country with lower savings. This is because higher savings mean higher BGP capital and, consequently (due to decreasing returns to this factor), lower MPK. Logically, this is translated into an inequality in real interest rates. The rate of return in the high-saving country is not half that of the low-saving country because MPK in the BGP is not linear in saving rates. However, the di erence is always positive. 4. Productivity di erences with international capital mobility. Consider two Solow economies with identical Cobb-Douglas production functions. Suppose that there is no technological change. Suppose also that the rate of depreciation of capital and that the level of labor e ciency is the same in both economies. Since the statement does not say anything in particular about the saving rates, we let them be di erent: s > s : Let all other parameters be equal across economies, as in the previous question. (a) Show that if capital moves to the country with the higher return (higher marginal product), both countries will have the same output per worker in the steady state. Assume initially that the economies are closed and have di erent levels of output per e ciency worker (not necessarily the steady-state values). We also assume that capital chases productivity di erentials and that the capital markets clear instantaneously. Therefore, we assume that right after the capital markets open, marginal productivity of capital is equalized in both economies after an initial capital ow. MP K = MP K ) f 0 (k + ) = f 0 (k ) Using the assumption that the technology is Cobb-Douglas, we get (k + ) = (k ) ) = k k 4

5 Because we characterized the capital markets to clear immediatly MP K = MP K holds at all times. Hence, from then on (and on the road to the steady state), k = k = k also holds. Note that this also implies that _ k = _ k = _k, otherwise, it would be possible for the values of k to be di erent accross economies. Thus, the modi ed laws of motion can be written as: _k = s f(k ) (n + )k + F and _ k = s f(k ) (n + )k F where F is a strictly non-negative capital ow, since economy always saves more and would tend to move faster towards the steady state equilibrium. The magnitude of the ow can be calculated using the previous equations: F = s s f(k) We can then nd the steady state to which both economies move jointly: _k = 0 ) s k (n + )k + s s k = 0 ) k ( s +s = )= n+ = k is: Plugging this in the production function, we nd that production per worker ( s +s Ay = A )= n+ Hence, both economies converge to the same output per worker (b) Will the result in (a) continue to hold if countries have di erent labor e ciency levels? From (a) we see that both economies always converge to the same level of capital per e ciency worker if they di er in the parameter s only. This equilibrium is not a ected by the economies having a di erent level of worker e ciency, A. Output per worker, nevertheless, will be di erent and the di erence in A (constant for each economy in this question) rescales output per worker. y i = A i ( s s )= n+ for i = ; 5

6 5. Rapid growth (convergence) could be due to rapid technological progress or rapid capital accumulation. After World War II, Germany, France, and Italy grew faster in terms of income per capita than the United States. This happened although the savings rate and the population growth rates in these countries were similar. Could this be explained by the destruction of physical capital during WWII? How would you use data to see whether your argument is right or wrong? We rely on the growth accounting framework to answer the question: = F (A; K; L) = AF (K; L) (Assume technology enters linearly for simplicity) _ = _ AF (K;L) + AF K K _ + AF L L _ A = _ A + AF K K _ K K + AF L L _ L L = T d F P + K ^K+L ^L where i represents the share of income that goes to factor i, T F P represents Total Factor Productivity and ^x denotes the growth rate of variable x. Finally, using the fact that with constant returns to scale techonology K + L = ; ^ ^L = d T F P + K ^K ^L This expression can be interpreted as saying that, in this framework, growth in income per capita has two possible sources: (i) growth of total factor productivity (improvements in the techniques used to aggregate inputs) and (ii) capital deepening (the excess of growth of capital stock over that of labor supply or, equivalently, growth of capital per capita). To focus initially on the e ect of physical capital only, we carry out the analysis under the following assumptions: (i) The same production function describes technology both in the US and in Europe (ii) Pre-war levels of e ciency and output per capita were the same in both regions As seen previously, the economic e ect of the war can be understood as a destruction of physical capital in per capita terms, without any alteration in the fundamentals of the economy. It can be seen that, in this scenario, growth in Europe needed to be greater than that of the US, strictly due to capital accumulation. That is, the destruction of capital triggers always some additional growth compared to steady state growth, given the parameters of the economy. But from the growth accounting equation, we see that growth of income per capita has two possible sources, only one of which is capital accumulation. 6

7 Hence, we must also see whether TFP was increasing signi cantly during this period and, if this were the case, which of the two sources was behind rapid European growth. Data should be used to assess this issue. In particular, we could replicate oung s estimations, but using the European region as a whole, and thus identify the relative importance of each source. (i) First, we could take the average values for the whole period (as seen in class, Europe stops converging, to a certain extent, around 990). This will give a rst indication of whether the nature of this growth was more "transitional" (i.e., due to capital accumulation) or "long-run" (i.e., due to technological progress). (ii) It would also be interesting to divide the sample, as oung did, in veor ten-year periods. This will allow us to see how the importance of each source varies in time. If the hypothesis that capital accumulation was the main factor is correct, we would observe a decrease in the importance of capital per capita accumulation as Europe approaches its steady state. TFP s behavior, on the other hand, would be di cult to predict. We would expect its relative importance to increase as we move to the latter years of the sample, because it is unlikely that Europe were able to develop technology too rapidly right after WWII. In fact, if our assumption of equal pre-war e ciency levels were correct, it would be unlikely to observe that T d F P EUR > T d F P US throughout the whole sample, as this would imply that Europe s overall level of e ciency surpassed that of the US. In conclusion, if the Solow model describes adequately this economies, physical capital destruction should induce faster growth in Europe. In fact, if it is true that European growth is mostly due to capital accumulation, we should observe that a signi cant part of it can be accounted for by capital deepening (most likely, for the earlier observations of the sample). However, if TFP growth also proved to be a source of growth, we could not argue that growth was uniquely due to capital deepening. As explained in the next question, both factors seem to have been important. 7

8 Figure y% = AL ( a δ ) + k % f ( k % ) s f ( k% ) k% = K AL k % k % BGP Figure y = L ( n δ ) + k ( n δ ) + k f ( k ) c c s f ( k ) k = K L k BGP k BGP

9 Figure 3 k BGP t y BGP t c BGP t