Textbook Solutions Ch.14
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CHAPTER 14 Classical Growth Theory This chapter presents the major growth theory model, known as the neoclassical, classical or Solow growth model. It focuses on the impact of long-run changes in labour, capital and the technology of production, as related by the production function. It omits the financial and government sectors from the analysis. The focus of this analysis is on the steady state, which is that long-run path of the economy along which, in the absence of technical change, the capital–labour ratio is constant. This chapter first presents the Solow model without technical change and then modifies it to incorporate technical change, which is shown to have been historically the main cause of the rise in the standards of living. The final part of this chapter presents the historical epochs of development, starting with the Malthusian one for primarily agricultural societies with land and labour as the main factors of production and without significant technical change over centuries.
14.1 Answers to Book Questions1 14.1.1 Review and discussion questions 1. Present the benchmark Solow growth model (including its assumptions) without technical change and show diagrammatically its steady-state (SS) position. Is this position stable? Explain your answer, using the appropriate diagram. Technology of production The Solow model assumes a production function of the form: Y = F(K, L)
YK , YL > 0; YKK , YLL < 0, YLK > 0
Since this production function is assumed to have constant returns to scale: αY = F(αK, αL) where α can be any positive constant. This equation asserts that if both labour and capital are increased in the same proportion (α), output will also increase in the same proportion. If we set α equal to 1/L, this equation implies that: Y/L = F(K/L, 1) which can be rewritten as: y = f(k)
yk > 0,
ykk < 0
1 Note: students’ answers do not have to include the parts that have been put in brackets of the type [ ].
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This form of the production function asserts that, for the given technology, output per worker depends only on capital per worker. Saving and investment The Solow model assumes that in the aggregate the average propensity to save σ is constant, and so the economy’s total saving S is: S = σY Per capita saving s is: s = S/L = σY/L = σy The capital stock K changes by the amount of real investment I. Since there is continuous equilibrium in the commodity market in this closed economy without a government sector, investment (I) and the change in the capital stock (K ) must equal saving S. Designating the change in K as K (= ∂K/∂t = change in K per period): K = I = S Hence, K = σY Labour force growth The labour force growth rate is assumed to be constant at n. Designating the growth rate of the labour force L by L , this assumption is: L =
1 ∂L L = =n L ∂t L
Analysis Note that k = K/L, which implies that: k = K − L Since k = k /k, K = K /K and L = n, we have: k /k = K /K − n Multiplying each term in this equation by k gives: k = K (k/K) − nk where K = S and k/K = 1/L. Hence, k = S/L − nk Since S = σY and Y/L = f(k), we get: k = σY/L − nk = σf(k) − nk The SS of the Solow model is defined as the long-run equilibrium with a constant capital–labour ratio. That is, in the SS: k = 0
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Output units
nk
f(k)
σ f(k)
k1
k*0
k2
k
Figure 14.1
which implies that: σf(k) = nk This is the fundamental equation (of the SS) of the Solow model. Diagrammatic analysis Figure 14.1 graphs the fundamental equation of the SS for the Solow model with k on the horizontal axis. The curve marked f(k) is output per capita. Since yk > 0 and ykk < 0, this curve is concave. σf(k) — which is also concave — measures the per capita availability of new capital (through saving), while the straight line representing nk measures the capital requirements of new workers at the existing capital–labour ratio. Since n is a constant, nk is represented by a straight line from the origin. The SS, with σf(k) = nk, occurs at k0∗ . To the left of k0∗ , at k1 , saving (i.e., new capital) is greater than required to equip new workers with capital at the existing capital–labour ratio and capital per worker increases for all workers, causing a rightward movement towards k0∗ . To the right of this point, say at k2 , saving (i.e., new capital) is less than required to equip new workers with capital at the existing capital– labour ratio. Since all workers have to have the identical amount of capital per worker, the capital provided to all workers will decrease, prompting a leftward movement to k0∗ . Therefore, if the economy is away from k0∗ , it will move back to k0∗ . That is, k0∗ is a stable SS capital–labour ratio. 2. Is the economy always in the steady state (SS)? If not, assuming that there is no technical change, what are the differences in the growth rates of (a) output and (b) of the standard of living between the SS position of the economy and the positions prior to reaching it? The economy is not always in the SS, it may be above or below the SS following shocks that change the capital–labour ratio (for example, after a war). If an economy is below SS, at a point like k1 , saving is greater than what is needed to equip new workers with capital so that the capital per worker increases. As a result, workers become more productive, which leads to higher growth rates of output and standard of living. 3. It is often claimed that (a) the growth rate of output, (b) the level of the standard of living and (c) the growth rate of the standard of living are negatively related to the population growth rate. Using two different labour force growth rates for illustration in your analysis, show whether each of the above relationships is implied by the Solow growth model without technical change? Give your answers for (i) the steady state and (ii) the pre-steady state.
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σ f(k)
k*0
k*1
k
Figure 14.3
Figure 14.3 shows the value of nk for two different labour growth rates, n0 > n1 and σf(k). (i) Steady state (a) The growth rate of output is given by Y0∗ = n0 and Y1∗ = n1 . From n0 > n1 , we have Y0∗ > Y1∗ . Hence, the growth rate of output and the population growth rate are positively related. (b) The standard of living is represented by output per capita. From Figure 14.3, it is clear that a lower population growth leads to a higher capital–labour ratio at the SS since k0∗ < k1∗ . In the SS, y∗ = f(k∗ ), where yk > 0 and ykk < 0. Hence, a higher SS per capita capital stock implies a higher standard of living: population growth rate and standard of living are negatively related. (c) In the SS, the growth rate of output per capita is zero, f (k∗ ) = 0, for any k∗ . Hence, in the SS, the growth rate of the standard of living is not related to the population growth rate, f (k0∗ ) = f (k1∗ ). (ii) Pre-steady state Assume that the economy was in SS0 (at k0∗ ) and that the population growth rate decreases to n1 . At k0∗ , the old SS, the growth rate of the per capita capital stock is now positive, i.e., k > 0. (a) In the pre-SS, the growth rate of output could increase or decrease. This is because the increase in the per capita capital stock increases the growth rate of output, while the decrease in the labour growth rate decreases the growth rate of output. The total effect on the growth rate of output is thus ambiguous. (b) In the pre-SS period, the decrease in the population growth rate increases the growth rate of k over time, which increases y each period. Hence, the standard of living each period is higher and is negatively related to the population growth rate. (c) In the pre-SS period, as the population growth rate decreases, few new workers are coming into the labour force, and so less of the new capital is required for the new workers. k rises for all workers. Hence, the growth rate of k increases above its previous SS value. This implies that the growth rate of y rises above its previous SS value. Hence, in the pre-SS period, the growth rate of the standard of living is negatively related to the population growth rate. 4. It is often claimed that the economy’s growth rate is independent of the saving rate of the economy. To investigate its implications in the Solow growth model without technical change, show the impact of increases and decreases in the saving rate on (i) the SS growth rate and (ii) the pre-SS growth rate.
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Output units
nk
σ 1 f(k) σ 0 f(k)
k*0
k*1
k
Figure 14.4
An increase in the marginal propensity to save (MPS) from σ0 to σ1 shifts the σf(k) curve upwards in Figure 14.4, from σ0 f(k) to σ1 f(k) and establishes a new SS at k1∗ . Since the capital per worker has increased to k1∗ , the SS output per worker — and therefore the standard of living — will be higher. However, the fundamental growth equation implies that the growth rate of output per worker in the new SS, as in the old one, will be zero. Therefore, in both the old and new steady states, output will grow at the labour force growth rate n. Note, however, that the output growth rate will rise above n during the transition from k0∗ to k1∗ . For a decrease in the saving rate, the Solow model shows that it reduces investment, the pre-SS growth rate and future living standards in both the transition to the new SS and the new SS itself — even though it does not change the SS growth rate of output per capita. Our analysis shows that: (a) In the pre-SS stage, ceteris paribus, countries with higher saving rates will have higher growth rates of output and of output per worker, and so their standards of living will improve at a faster rate. (b) In the SS, ceteris paribus, countries with higher saving rates will have the same growth rate (zero) of output per worker as those with lower saving rates. (c) In the SS, ceteris paribus, countries with higher saving rates will have higher levels of output per worker and, therefore, higher standards of living. 5. It is sometimes claimed that while a war (on one’s own land) destroys a great deal of the economy’s capital, it also leads to very significant inventions and innovations that are incorporated into civilian production after the war. Assuming that the economy was in the steady state (SS) at the start of the war, present its diagrammatic analysis for the Solow-type economy, comparing the pre- and post-war SS output per capita. What happens in the ‘recovery’ period during which the capital stock is still below the pre-war SS level? This question is answered by first considering a decrease in the capital stock, with no change in technology. (A) First, assume that the economy was initially in the SS at k0∗ and that the production function remains unchanged but some capital is destroyed by the war. As a result of the war, the per capita capital stock of the economy falls to k1 . At k1 , k > 0 and y > 0. The economy would eventually return to the old SS. In this SS, the per capita capital stock will again become k0∗ , and k = y = 0.
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nk
σ f1(k) σ f0(k)
k1
k*0
k*2
k
Figure 14.5
(B) Now assume that while the capital stock per worker fell to k1 , the inventions and innovations increased the productivity of labour such that the σf(k) curve shifts up from f0 (k) to f1 (k). At k1 , the growth rates of k and y become higher than under (A). They are also higher than at the old SS values of k0∗ and y0∗ . The economy’s growth rates of k and y increase and remain higher than under the old production function, until the new SS is reached at k2∗ . In this new SS, per capita capital and income are higher than at the start of the war. 6. ‘Countries (including the LDCs) in which governments pursued plans to increase their saving rates as a means of boosting their growth rates were misguided’. Discuss. This statement is correct only if the steady states before and after the increase in the saving rate are compared, since in both cases the SS growth rate of output equals the labour force growth rate, which is assumed to be unchanged. However, if we compare pre-SS growth rates, the statement is not correct: the higher saving rate increases the growth rates of capital and output per worker. This will occur whether the economy was initially in an SS or in a pre-SS stage. An additional consideration becomes relevant for the LDCs if the economy has a production function with an initial segment representing increasing MPK and the economy had a capital–labour ratio less than the take-off one. For this case, consider the production function shown in Figure 14.6, and consider an increase in the saving rate from σ0 to σ1 . The SS at k0∗ is the take-off capital–labour ratio: to the right of it, the economy takes off to higher standards of living. Now assume two economies, economy 1 with capital–labour ratio k1 and economy 2 with capital–labour ratio k2 . After the rise in the saving rate, whereas economy 2 takes off to higher standards of living (ultimately reaching k2∗ ), the capital–labour ratio of economy 1 will reach only the new take-off ratio k1∗ . Therefore, for LDCs with a capital–labour ratio below the take-off one, the rise in the saving rate can make a vital difference to the growth rate depending upon the initial state of the economy. 7. What is meant by ‘convergence’? How does the Solow model explain convergence? ‘Convergence’ means that different countries eventually achieve the same output per capita. The standard version of the Solow model with diminishing MPK implies that all countries would converge to the same
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nk
σ 1 f(k) σ 0 f(k)
k1 k*1 k2 k*0
k*2
Figure 14.6
SS k and y ratios if they have: (a) the same production function with a single technology possessing diminishing marginal productivity of capital; (b) the same saving propensity; (c) the same population growth rate. If countries have different production functions, saving propensities and labour force growth rates, the Solow model implies that capital per worker and output per worker would become constant in their respective steady states, though at different levels. In this case, standards of living will not converge. 8. According to the Solow model, what factors determine the steady-state (SS) growth of output per capita? Can the long-run growth rate of output differ from the SS one? Can the short-run growth rate of output differ from the long-run and the SS ones? Explain your answers. According to the benchmark Solow model without technical change, the SS capital–labour ratio is constant and so is the output–labour ratio. Therefore, in the SS, the growth rate of output per capita is zero. Note that if the model includes technical change, the SS growth rate of output per capita will be A , the growth of total factor productivity. The SS, along which the capital–labour ratio does not change, is a special case of a long-run growth path. Therefore, the long-run growth rate can differ from the SS one. For example, in the benchmark Solow model, in the pre-SS phase, the capital–labour ratio increases, even though the economy is moving along a long-run growth path. The Solow model assumes the economy to be in long-run full employment all the time, and so the study of the actual growth path, e.g., over a business cycle, is omitted. However, in reality, the economy may have actual growth rates different from the long-run and SS ones. 9. How does the growth accounting method calculate the contribution (to the host economy) of immigrants? What does the Solow model imply for the impact of a very significant rate of increase in immigration on the steady state? The growth accounting method calculates the contribution to the host economy of immigrants by their marginal productivity of labour (assumed to be equal to their real wage) multiplied by their number. A very significant increase in the immigration growth rate is represented in the Solow model by an increase in the population growth rate, as shown in Figure 14.9. An increase in the population growth
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Macroeconomics: Answers to BOOK QS CH 14 Output units n1k n0k
σ f(k)
k*1
k*0
k
Figure 14.9
from n0 to n1 turns the nk-locus upwards. With the higher population growth rate, the capital–labour ratio decreases from k0∗ to k1∗ . There is a corresponding decrease in the level of the SS output per capita. 10. Present the Malthusian theory of growth. What are its critical assumptions that cause long-run living standards to remain at the subsistence level? Are there countries for which this analysis is still applicable or at least indicative of their growth pattern? The assumptions of the Malthusian growth theory are: (i) The factors of production are labour and land. Of these, land is fixed in supply, and so the marginal productivity of labour, the only variable input in this theory, diminishes. (ii) Population is positively related to the standard of living, and grows faster than productivity per worker in the economy. This occurs through rising birth rates and/or falling mortality rates due to the improvements in living standards. (iii) Technology does not change. The Malthusian theory of growth states that the long-run standard of living fluctuates around the subsistence level. If standards of living rise above the subsistence level, population will increase faster than output, and so the standards of living will fall to the subsistence level. If standards of living fall below the subsistence level, population will decrease relative to output, and so the standards of living will rise to the subsistence level. Therefore, in this theory, the population growth rate adjusts to drive the economy to the SS at the subsistence level. [The Malthusian growth theory was appropriate for the pre-industrial societies dominated by agriculture and without continuing technical improvements of macroeconomic relevance. In these societies, the capital–labour ratio was quite low and did not change much over long periods, and so the explicit treatment of capital was not necessary. The emphasis on land was justified since agriculture was the dominant economic activity.] A country can be in the pre-Malthusian stage if it is dominated by agriculture, its industry is in the pre-industrialisation state and it is unable to innovate. Some LDCs currently do seem to fall in this category. 11. Discuss the relationship (or different types of relationships) that seems to have occurred historically between population growth rates and living standards.
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The historical relationship between population and growth rates can be divided into the following distinct periods: The Malthusian stage: With no technical change, living standards were virtually constant around the SS. An increase in standards of living would result in an increase in population, and so the initial improvement in living standards is eventually cancelled out. The post-Malthusian stage: The introduction of technical change leads to an increase in output and living standards. With better food, sanitation, etc., population increases significantly in response to these changes. Because of the technical improvements, this increase in population and living standards continues. The modern growth stage: This stage has fairly high and rising living standards. Medical advances are a major factor in this stage. Technical change, notably birth control technology, leads to a decrease in the growth rate of population due to decreases in the number of children per woman while living standards keep improving. This effect dominates the increase in the population growth rate due to increases in longevity brought about by improvements in medical technology. The post-modern growth stage: This stage is speculative at present. In this stage, the number of children per woman stabilises at less than two but longevity due to medical technology keeps rising, and so there may be some decline in the population. Note that globalisation has accelerated the spread of technology, and so the less developed economies can now move through these various phases in a shorter time frame.
14.2 Advanced and Technical Questions T1. Assume that labour’s and capital’s shares of output are respectively 0.7 and 0.3, and that output is growing at 3% annually. There is no technical change. What is the growth rate of total factor productivity if: (i) Both labour and capital are growing at 2%? Growth of output due to capital growth = αK K = 0.3(0.02) = 0.006 = 0.6% Growth of output due to labour force growth = αL L = 0.7(0.02) = 0.014 = 1.4% Growth of output due to technical change = A (T) = 0.03 − 0.006 − 0.014 = 0.01 = 1% Hence, the total output growth of 3% can be decomposed as follows: 1% each year comes from improvements in technology, 0.6% from the growth of capital and 1.4% from the increase in labour. (ii) Labour is stationary while capital is growing at 3%? Growth of output due to capital growth = αK K = 0.3(0.03) = 0.009 = 0.9% Growth of output due to labour force growth = αL L = 0.7(0.00) = 0 Growth of output due to technical change = A (T) = 0.03 − 0.009 = 0.021 = 2.1% T2. Suppose that increases in the labour force participation rate occur over a particular decade, after which the participation rate stabilises. Discuss its effects on the growth rate of output per capita over the decade relative to those before and after it. Specify any assumptions that you consider to be relevant to your analysis. This sort of increase can be interpreted as a temporary increase in the growth rate of labour. In this case, during the decade, the relevant line is n1 k while it is n0 k before and after the decade of increases in the participation rate. Starting with the initial SS position shown in Figure 14.T2 as k0∗ , the incorporation of a sudden increase in the labour force participation rate would lead to an increase in n (the labour force growth rate) and
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n1k n0k
σ f(k)
k*1
k*0
k
Figure 14.T2
would lower k below k0∗ to k1 . This lower capital–labour ratio decreases output per worker (thereby lowering the standard of living) below its SS value. This reduces output and real wages per worker, but there are more income earners per family. However, after the decade of increases in the participation rate, the labour force growth rate goes back to its original level n0 . At k1 , σf(k) > n0 k, and so k starts increasing over time. This increase in k increases output per worker. Gradually, the economy’s capital–labour ratio returns to k0∗ , at which time the economy returns to its old SS. T3. What explanations have been offered for the falling population growth rates in the modern period? Can it be offset and how? Is there a limit to the fall in the population of (a) individual countries and (b) the world? Discuss. The explanation given for a fall in the birth rates in the modern period emphasises a decrease in the desired number of children per woman. This desired rate can be implemented by the easily available birth control technology. The desired number of children per family has fallen partly because of the advancing technology, which raised the return to human capital and induced increased investment in the education of (fewer) children. Another factor has been the increasing participation of women in the labour force — which substantially increased the opportunity cost of having and raising children. Other causes for this fall include changes in the social and cultural patterns affecting the family, and its socially desirable size. If the shift in social preferences is not affected by economic considerations, the fall in the birth rates cannot be significantly offset by economic incentives to have more children. However, if the fall in the birth rates is the result of increasing costs of raising children, then economic incentives can compensate for these expenses and reverse the downward trend in birth rates. (a) For individual countries, the decline in the birth rates can be offset by immigration. Adequate immigration can prevent a fall in the population growth rate. (b) For the world, there is no limit to the fall in population if births remain below deaths. T4. Assuming that all countries in the world have entered the ‘modern growth stage’ (as defined in this chapter), present the analysis of the long-run steady state for the world population and its standards of living under the modern growth theory with exogenous labour-saving technical change. The modern growth stage opens with living standards that are already substantially above the subsistence level. In this stage, the population growth rate is now negatively related to the (further) improvements in the standard of living. Mortality has already declined considerably prior to this stage and does not
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Output units
(n + η)kE
σ f(kE)
kE1
kE*0
kE
Figure 14.T4
significantly fall further. But birth rates fall due to a decrease in the desired number of children per woman, which become feasible through an easily available low-cost birth control technology. Therefore, in the long run, the world population will decline. Hence, the labour force growth rate will be negative. The analysis of this stage can be based on the Solow model with the addition of the above assumptions of a negative labour force growth rate and positive labour-saving technical change. The following answer assumes that the sum of the labour growth rate and the rate of growth of labour efficiency, (n + η), is positive. The Solow model then implies that the improvements in the standard of living in the pre-SS stage will occur due to both falling population growth rates — which raise the growth rate of capital per worker — and technical change. In the SS, the standards of living will grow at the rate η. The relevant diagram is shown in Figure 14.T4. In this diagram, k1E illustrates the pre-SS position, with both increases in the capital–labour ratio and technical change. Living standards rise due to both these changes. k0E is the SS position with a constant KE /L ratio, with increases in living standards at the rate η due only to technical change. Basic growth model for this chapter: Y/L = 100(K/L) − 0.5(K/L)2 APC = MPC = 0.8 L = 0.04 T5. For the basic growth model of this chapter, calculate the marginal product of capital per worker (MPk). Is MPk constant in the steady state of the basic Solow model without technical change? Explain your answer. The production function is: y = 100K/L − 0.5(K/L)2 = 100k − 0.5k2 The marginal product of capital per worker (MPk) is: MPk = ∂y/∂k = 100 − k In the SS of the Solow model without technical change, MPk is constant because k is constant.
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T6. Using the basic growth model of this chapter, (i) What is the SS value of k? We know that σ = 1 − MPC = 1 − 0.8 = 0.2, that n = L = 0.04 and that f(k) = 100k − 0.5k2 . Substituting these values in the fundamental SS equation [σf(k) = nk] of the Solow model: 0.2(100k − 0.5k2 ) = 0.04k 20k − 0.1k2 = 0.04k 20 − 0.1k = 0.04 k = 199.6 (ii) What is the SS value of y? The SS value of y is given by: y = f(k) = 100k − 0.5k2 = 100(199.6) − 0.5(199.6)2 = 39.92 The level of saving per capita in the SS is given by: σf(k) = 0.2(39.92) = 7.984 (iii) What are the steady-state levels of (a) consumption per capita and (b) saving per capita? The level of consumption per capita can be found as follows: c = APC · f(k) = 0.8(39.92) = 31.936 The level of saving per capita in the SS is given by: σf(k) = 0.2(39.92) = 7.984 (iv) If the MPC changes to 0.7, recalculate your answers to the previous questions. We know that σ = 1 − MPC = 1 − 0.7 = 0.3, that n = L = 0.04 and that f(k) = 100k − 0.5k2 . Substituting these values in the fundamental SS equation of the Solow model: 0.3(100k − 0.5k2 ) = 0.04k 30k − 0.15k2 = 0.04k 30 − 0.15k = 0.04 k = 199.73 The SS value of y is given by: y = f(k) = 100k − 0.5k2 = 100(199.73) − 0.5(199.73)2 = 26.96 The level of saving per capita in the SS is given by: σf(k) = 0.3(26.96) = 8.09 Consumption is: c = APC · f(k) = 0.7(26.96) = 18.87 (v) What is the change in the SS growth rates of output and capital for the two values of MPC? There is no change in the SS growth rates of either output or capital. For both values of MPC, output and capital grow at the labour force growth rate, i.e., n = 0.04. Revised growth model A for this chapter, with labour-saving technical change: yE = 1,000kE − 2kE2 APC = MPC = 0.8 L = n = 0.02 and η = 0.02 [n is the growth rate of workers; η is the rate of growth of efficiency per worker.]
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T7. Using the revised growth model A of this chapter, (i) What are the SS growth rates of kE and k? (ii) What are the SS growth rates of yE and y? This question can be answered without using the information on the production function and the APC/MPC. Only the information of η = 0.02 is needed. The answer is as follows: yE = kE = 0 y = k = η = 0.02 Revised growth model B for this chapter, with labour-saving technical change: yE = 2,000kE − 4kE2 APC = MPC = 0.9 E = n + η n = 0.05, η = 0.015 T8. Using the revised model B of this chapter, (i) What is the SS value of kE ? To answer this question, we start with the fundamental SS equation for the Solow model with technical change: σyE = (n + η)kE We know n and η, and can compute σ = 1 − APC = 1 − 0.8 = 0.2. Therefore, 0.2(2,000kE − 4kE2 ) = (0.05 + 0.015)kE Dividing each side by kE : 400 − 0.8kE = 0.065 kE = 499.92 (ii) What is the SS value of yE ? The value of yE can be computed from the production function: yE = 2,000kE − 4kE2 = 2,000(499.92) − 4(499.92)2 = 159.97 (iii) What is the growth rate of consumption per capita in the steady state? Since consumption is a fixed fraction of output, it will grow at the same rate as output. At the SS, output growth will be equal to the growth rate of efficiency per worker η = 0.015. Therefore, the growth rate of consumption per capita will be 0.015. (iv) Suppose that because of a low birth rate, aging population with rising numbers of retirees and inadequate immigration levels, the labour force growth rate falls to −0.01, while η remains at 0.015. Recalculate your answers to (a) to (c). To answer this question, we start with the fundamental SS equation for the Solow model with technical change: σyE = (n + η)kE We know n and η, and can compute σ = 1 − APC = 1 − 0.8 = 0.2. Therefore, 0.2(2,000kE − 4kE2 ) = (0.01 + 0.015)kE Dividing each side by kE : 400 − 0.8kE = 0.025
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(n0 + η 0)kE (n1 + η 0)kE
σ f(kE)
kE*0
kE*1
kE
Figure 14.T8
kE = 499.97 The value of yE can be computed from the production function: yE = 2,000kE − 4kE2 = 2,000(499.97) − 4(499.97)2 = 60.00 Note that the SS growth rate of consumption per capita will remain the same at 0.015. Use a diagram to illustrate the general nature of your results for this part of the question. In Figure 14.T8, the decrease in the labour force growth rate would be represented by a downward shift/rotation of the (flatter) nkE line, while an increase in rate of labour-saving technical change would be represented by an upward shift of the nkE line. Assuming that the rate of technical change remains constant at η0 , a decrease in the labour force growth rate from n0 to n1 rotates the (n + η)kE line down from (n0 + η0 )kE to (n1 + η0 )kE , and so the lower labour force growth rate results in a higher SS level of capital per worker, which would raise the SS output per worker.
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CHAPTER 14 Classical Growth Theory This chapter presents the major growth theory model, known as the neoclassical, classical or Solow growth model. It focuses on the impact of long-run changes in labour, capital and the technology of production, as related by the production function. It omits the financial and government sectors from the analysis. The focus of this analysis is on the steady state, which is that long-run path of the economy along which, in the absence of technical change, the capital–labour ratio is constant. This chapter first presents the Solow model without technical change and then modifies it to incorporate technical change, which is shown to have been historically the main cause of the rise in the standards of living. The final part of this chapter presents the historical epochs of development, starting with the Malthusian one for primarily agricultural societies with land and labour as the main factors of production and without significant technical change over centuries.
14.1 Answers to Book Questions1 14.1.1 Review and discussion questions 1. Present the benchmark Solow growth model (including its assumptions) without technical change and show diagrammatically its steady-state (SS) position. Is this position stable? Explain your answer, using the appropriate diagram. Technology of production The Solow model assumes a production function of the form: Y = F(K, L)
YK , YL > 0; YKK , YLL < 0, YLK > 0
Since this production function is assumed to have constant returns to scale: αY = F(αK, αL) where α can be any positive constant. This equation asserts that if both labour and capital are increased in the same proportion (α), output will also increase in the same proportion. If we set α equal to 1/L, this equation implies that: Y/L = F(K/L, 1) which can be rewritten as: y = f(k)
yk > 0,
ykk < 0
1 Note: students’ answers do not have to include the parts that have been put in brackets of the type [ ].
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This form of the production function asserts that, for the given technology, output per worker depends only on capital per worker. Saving and investment The Solow model assumes that in the aggregate the average propensity to save σ is constant, and so the economy’s total saving S is: S = σY Per capita saving s is: s = S/L = σY/L = σy The capital stock K changes by the amount of real investment I. Since there is continuous equilibrium in the commodity market in this closed economy without a government sector, investment (I) and the change in the capital stock (K ) must equal saving S. Designating the change in K as K (= ∂K/∂t = change in K per period): K = I = S Hence, K = σY Labour force growth The labour force growth rate is assumed to be constant at n. Designating the growth rate of the labour force L by L , this assumption is: L =
1 ∂L L = =n L ∂t L
Analysis Note that k = K/L, which implies that: k = K − L Since k = k /k, K = K /K and L = n, we have: k /k = K /K − n Multiplying each term in this equation by k gives: k = K (k/K) − nk where K = S and k/K = 1/L. Hence, k = S/L − nk Since S = σY and Y/L = f(k), we get: k = σY/L − nk = σf(k) − nk The SS of the Solow model is defined as the long-run equilibrium with a constant capital–labour ratio. That is, in the SS: k = 0
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Output units
nk
f(k)
σ f(k)
k1
k*0
k2
k
Figure 14.1
which implies that: σf(k) = nk This is the fundamental equation (of the SS) of the Solow model. Diagrammatic analysis Figure 14.1 graphs the fundamental equation of the SS for the Solow model with k on the horizontal axis. The curve marked f(k) is output per capita. Since yk > 0 and ykk < 0, this curve is concave. σf(k) — which is also concave — measures the per capita availability of new capital (through saving), while the straight line representing nk measures the capital requirements of new workers at the existing capital–labour ratio. Since n is a constant, nk is represented by a straight line from the origin. The SS, with σf(k) = nk, occurs at k0∗ . To the left of k0∗ , at k1 , saving (i.e., new capital) is greater than required to equip new workers with capital at the existing capital–labour ratio and capital per worker increases for all workers, causing a rightward movement towards k0∗ . To the right of this point, say at k2 , saving (i.e., new capital) is less than required to equip new workers with capital at the existing capital– labour ratio. Since all workers have to have the identical amount of capital per worker, the capital provided to all workers will decrease, prompting a leftward movement to k0∗ . Therefore, if the economy is away from k0∗ , it will move back to k0∗ . That is, k0∗ is a stable SS capital–labour ratio. 2. Is the economy always in the steady state (SS)? If not, assuming that there is no technical change, what are the differences in the growth rates of (a) output and (b) of the standard of living between the SS position of the economy and the positions prior to reaching it? The economy is not always in the SS, it may be above or below the SS following shocks that change the capital–labour ratio (for example, after a war). If an economy is below SS, at a point like k1 , saving is greater than what is needed to equip new workers with capital so that the capital per worker increases. As a result, workers become more productive, which leads to higher growth rates of output and standard of living. 3. It is often claimed that (a) the growth rate of output, (b) the level of the standard of living and (c) the growth rate of the standard of living are negatively related to the population growth rate. Using two different labour force growth rates for illustration in your analysis, show whether each of the above relationships is implied by the Solow growth model without technical change? Give your answers for (i) the steady state and (ii) the pre-steady state.
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σ f(k)
k*0
k*1
k
Figure 14.3
Figure 14.3 shows the value of nk for two different labour growth rates, n0 > n1 and σf(k). (i) Steady state (a) The growth rate of output is given by Y0∗ = n0 and Y1∗ = n1 . From n0 > n1 , we have Y0∗ > Y1∗ . Hence, the growth rate of output and the population growth rate are positively related. (b) The standard of living is represented by output per capita. From Figure 14.3, it is clear that a lower population growth leads to a higher capital–labour ratio at the SS since k0∗ < k1∗ . In the SS, y∗ = f(k∗ ), where yk > 0 and ykk < 0. Hence, a higher SS per capita capital stock implies a higher standard of living: population growth rate and standard of living are negatively related. (c) In the SS, the growth rate of output per capita is zero, f (k∗ ) = 0, for any k∗ . Hence, in the SS, the growth rate of the standard of living is not related to the population growth rate, f (k0∗ ) = f (k1∗ ). (ii) Pre-steady state Assume that the economy was in SS0 (at k0∗ ) and that the population growth rate decreases to n1 . At k0∗ , the old SS, the growth rate of the per capita capital stock is now positive, i.e., k > 0. (a) In the pre-SS, the growth rate of output could increase or decrease. This is because the increase in the per capita capital stock increases the growth rate of output, while the decrease in the labour growth rate decreases the growth rate of output. The total effect on the growth rate of output is thus ambiguous. (b) In the pre-SS period, the decrease in the population growth rate increases the growth rate of k over time, which increases y each period. Hence, the standard of living each period is higher and is negatively related to the population growth rate. (c) In the pre-SS period, as the population growth rate decreases, few new workers are coming into the labour force, and so less of the new capital is required for the new workers. k rises for all workers. Hence, the growth rate of k increases above its previous SS value. This implies that the growth rate of y rises above its previous SS value. Hence, in the pre-SS period, the growth rate of the standard of living is negatively related to the population growth rate. 4. It is often claimed that the economy’s growth rate is independent of the saving rate of the economy. To investigate its implications in the Solow growth model without technical change, show the impact of increases and decreases in the saving rate on (i) the SS growth rate and (ii) the pre-SS growth rate.
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Output units
nk
σ 1 f(k) σ 0 f(k)
k*0
k*1
k
Figure 14.4
An increase in the marginal propensity to save (MPS) from σ0 to σ1 shifts the σf(k) curve upwards in Figure 14.4, from σ0 f(k) to σ1 f(k) and establishes a new SS at k1∗ . Since the capital per worker has increased to k1∗ , the SS output per worker — and therefore the standard of living — will be higher. However, the fundamental growth equation implies that the growth rate of output per worker in the new SS, as in the old one, will be zero. Therefore, in both the old and new steady states, output will grow at the labour force growth rate n. Note, however, that the output growth rate will rise above n during the transition from k0∗ to k1∗ . For a decrease in the saving rate, the Solow model shows that it reduces investment, the pre-SS growth rate and future living standards in both the transition to the new SS and the new SS itself — even though it does not change the SS growth rate of output per capita. Our analysis shows that: (a) In the pre-SS stage, ceteris paribus, countries with higher saving rates will have higher growth rates of output and of output per worker, and so their standards of living will improve at a faster rate. (b) In the SS, ceteris paribus, countries with higher saving rates will have the same growth rate (zero) of output per worker as those with lower saving rates. (c) In the SS, ceteris paribus, countries with higher saving rates will have higher levels of output per worker and, therefore, higher standards of living. 5. It is sometimes claimed that while a war (on one’s own land) destroys a great deal of the economy’s capital, it also leads to very significant inventions and innovations that are incorporated into civilian production after the war. Assuming that the economy was in the steady state (SS) at the start of the war, present its diagrammatic analysis for the Solow-type economy, comparing the pre- and post-war SS output per capita. What happens in the ‘recovery’ period during which the capital stock is still below the pre-war SS level? This question is answered by first considering a decrease in the capital stock, with no change in technology. (A) First, assume that the economy was initially in the SS at k0∗ and that the production function remains unchanged but some capital is destroyed by the war. As a result of the war, the per capita capital stock of the economy falls to k1 . At k1 , k > 0 and y > 0. The economy would eventually return to the old SS. In this SS, the per capita capital stock will again become k0∗ , and k = y = 0.
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nk
σ f1(k) σ f0(k)
k1
k*0
k*2
k
Figure 14.5
(B) Now assume that while the capital stock per worker fell to k1 , the inventions and innovations increased the productivity of labour such that the σf(k) curve shifts up from f0 (k) to f1 (k). At k1 , the growth rates of k and y become higher than under (A). They are also higher than at the old SS values of k0∗ and y0∗ . The economy’s growth rates of k and y increase and remain higher than under the old production function, until the new SS is reached at k2∗ . In this new SS, per capita capital and income are higher than at the start of the war. 6. ‘Countries (including the LDCs) in which governments pursued plans to increase their saving rates as a means of boosting their growth rates were misguided’. Discuss. This statement is correct only if the steady states before and after the increase in the saving rate are compared, since in both cases the SS growth rate of output equals the labour force growth rate, which is assumed to be unchanged. However, if we compare pre-SS growth rates, the statement is not correct: the higher saving rate increases the growth rates of capital and output per worker. This will occur whether the economy was initially in an SS or in a pre-SS stage. An additional consideration becomes relevant for the LDCs if the economy has a production function with an initial segment representing increasing MPK and the economy had a capital–labour ratio less than the take-off one. For this case, consider the production function shown in Figure 14.6, and consider an increase in the saving rate from σ0 to σ1 . The SS at k0∗ is the take-off capital–labour ratio: to the right of it, the economy takes off to higher standards of living. Now assume two economies, economy 1 with capital–labour ratio k1 and economy 2 with capital–labour ratio k2 . After the rise in the saving rate, whereas economy 2 takes off to higher standards of living (ultimately reaching k2∗ ), the capital–labour ratio of economy 1 will reach only the new take-off ratio k1∗ . Therefore, for LDCs with a capital–labour ratio below the take-off one, the rise in the saving rate can make a vital difference to the growth rate depending upon the initial state of the economy. 7. What is meant by ‘convergence’? How does the Solow model explain convergence? ‘Convergence’ means that different countries eventually achieve the same output per capita. The standard version of the Solow model with diminishing MPK implies that all countries would converge to the same
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nk
σ 1 f(k) σ 0 f(k)
k1 k*1 k2 k*0
k*2
Figure 14.6
SS k and y ratios if they have: (a) the same production function with a single technology possessing diminishing marginal productivity of capital; (b) the same saving propensity; (c) the same population growth rate. If countries have different production functions, saving propensities and labour force growth rates, the Solow model implies that capital per worker and output per worker would become constant in their respective steady states, though at different levels. In this case, standards of living will not converge. 8. According to the Solow model, what factors determine the steady-state (SS) growth of output per capita? Can the long-run growth rate of output differ from the SS one? Can the short-run growth rate of output differ from the long-run and the SS ones? Explain your answers. According to the benchmark Solow model without technical change, the SS capital–labour ratio is constant and so is the output–labour ratio. Therefore, in the SS, the growth rate of output per capita is zero. Note that if the model includes technical change, the SS growth rate of output per capita will be A , the growth of total factor productivity. The SS, along which the capital–labour ratio does not change, is a special case of a long-run growth path. Therefore, the long-run growth rate can differ from the SS one. For example, in the benchmark Solow model, in the pre-SS phase, the capital–labour ratio increases, even though the economy is moving along a long-run growth path. The Solow model assumes the economy to be in long-run full employment all the time, and so the study of the actual growth path, e.g., over a business cycle, is omitted. However, in reality, the economy may have actual growth rates different from the long-run and SS ones. 9. How does the growth accounting method calculate the contribution (to the host economy) of immigrants? What does the Solow model imply for the impact of a very significant rate of increase in immigration on the steady state? The growth accounting method calculates the contribution to the host economy of immigrants by their marginal productivity of labour (assumed to be equal to their real wage) multiplied by their number. A very significant increase in the immigration growth rate is represented in the Solow model by an increase in the population growth rate, as shown in Figure 14.9. An increase in the population growth
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σ f(k)
k*1
k*0
k
Figure 14.9
from n0 to n1 turns the nk-locus upwards. With the higher population growth rate, the capital–labour ratio decreases from k0∗ to k1∗ . There is a corresponding decrease in the level of the SS output per capita. 10. Present the Malthusian theory of growth. What are its critical assumptions that cause long-run living standards to remain at the subsistence level? Are there countries for which this analysis is still applicable or at least indicative of their growth pattern? The assumptions of the Malthusian growth theory are: (i) The factors of production are labour and land. Of these, land is fixed in supply, and so the marginal productivity of labour, the only variable input in this theory, diminishes. (ii) Population is positively related to the standard of living, and grows faster than productivity per worker in the economy. This occurs through rising birth rates and/or falling mortality rates due to the improvements in living standards. (iii) Technology does not change. The Malthusian theory of growth states that the long-run standard of living fluctuates around the subsistence level. If standards of living rise above the subsistence level, population will increase faster than output, and so the standards of living will fall to the subsistence level. If standards of living fall below the subsistence level, population will decrease relative to output, and so the standards of living will rise to the subsistence level. Therefore, in this theory, the population growth rate adjusts to drive the economy to the SS at the subsistence level. [The Malthusian growth theory was appropriate for the pre-industrial societies dominated by agriculture and without continuing technical improvements of macroeconomic relevance. In these societies, the capital–labour ratio was quite low and did not change much over long periods, and so the explicit treatment of capital was not necessary. The emphasis on land was justified since agriculture was the dominant economic activity.] A country can be in the pre-Malthusian stage if it is dominated by agriculture, its industry is in the pre-industrialisation state and it is unable to innovate. Some LDCs currently do seem to fall in this category. 11. Discuss the relationship (or different types of relationships) that seems to have occurred historically between population growth rates and living standards.
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The historical relationship between population and growth rates can be divided into the following distinct periods: The Malthusian stage: With no technical change, living standards were virtually constant around the SS. An increase in standards of living would result in an increase in population, and so the initial improvement in living standards is eventually cancelled out. The post-Malthusian stage: The introduction of technical change leads to an increase in output and living standards. With better food, sanitation, etc., population increases significantly in response to these changes. Because of the technical improvements, this increase in population and living standards continues. The modern growth stage: This stage has fairly high and rising living standards. Medical advances are a major factor in this stage. Technical change, notably birth control technology, leads to a decrease in the growth rate of population due to decreases in the number of children per woman while living standards keep improving. This effect dominates the increase in the population growth rate due to increases in longevity brought about by improvements in medical technology. The post-modern growth stage: This stage is speculative at present. In this stage, the number of children per woman stabilises at less than two but longevity due to medical technology keeps rising, and so there may be some decline in the population. Note that globalisation has accelerated the spread of technology, and so the less developed economies can now move through these various phases in a shorter time frame.
14.2 Advanced and Technical Questions T1. Assume that labour’s and capital’s shares of output are respectively 0.7 and 0.3, and that output is growing at 3% annually. There is no technical change. What is the growth rate of total factor productivity if: (i) Both labour and capital are growing at 2%? Growth of output due to capital growth = αK K = 0.3(0.02) = 0.006 = 0.6% Growth of output due to labour force growth = αL L = 0.7(0.02) = 0.014 = 1.4% Growth of output due to technical change = A (T) = 0.03 − 0.006 − 0.014 = 0.01 = 1% Hence, the total output growth of 3% can be decomposed as follows: 1% each year comes from improvements in technology, 0.6% from the growth of capital and 1.4% from the increase in labour. (ii) Labour is stationary while capital is growing at 3%? Growth of output due to capital growth = αK K = 0.3(0.03) = 0.009 = 0.9% Growth of output due to labour force growth = αL L = 0.7(0.00) = 0 Growth of output due to technical change = A (T) = 0.03 − 0.009 = 0.021 = 2.1% T2. Suppose that increases in the labour force participation rate occur over a particular decade, after which the participation rate stabilises. Discuss its effects on the growth rate of output per capita over the decade relative to those before and after it. Specify any assumptions that you consider to be relevant to your analysis. This sort of increase can be interpreted as a temporary increase in the growth rate of labour. In this case, during the decade, the relevant line is n1 k while it is n0 k before and after the decade of increases in the participation rate. Starting with the initial SS position shown in Figure 14.T2 as k0∗ , the incorporation of a sudden increase in the labour force participation rate would lead to an increase in n (the labour force growth rate) and
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n1k n0k
σ f(k)
k*1
k*0
k
Figure 14.T2
would lower k below k0∗ to k1 . This lower capital–labour ratio decreases output per worker (thereby lowering the standard of living) below its SS value. This reduces output and real wages per worker, but there are more income earners per family. However, after the decade of increases in the participation rate, the labour force growth rate goes back to its original level n0 . At k1 , σf(k) > n0 k, and so k starts increasing over time. This increase in k increases output per worker. Gradually, the economy’s capital–labour ratio returns to k0∗ , at which time the economy returns to its old SS. T3. What explanations have been offered for the falling population growth rates in the modern period? Can it be offset and how? Is there a limit to the fall in the population of (a) individual countries and (b) the world? Discuss. The explanation given for a fall in the birth rates in the modern period emphasises a decrease in the desired number of children per woman. This desired rate can be implemented by the easily available birth control technology. The desired number of children per family has fallen partly because of the advancing technology, which raised the return to human capital and induced increased investment in the education of (fewer) children. Another factor has been the increasing participation of women in the labour force — which substantially increased the opportunity cost of having and raising children. Other causes for this fall include changes in the social and cultural patterns affecting the family, and its socially desirable size. If the shift in social preferences is not affected by economic considerations, the fall in the birth rates cannot be significantly offset by economic incentives to have more children. However, if the fall in the birth rates is the result of increasing costs of raising children, then economic incentives can compensate for these expenses and reverse the downward trend in birth rates. (a) For individual countries, the decline in the birth rates can be offset by immigration. Adequate immigration can prevent a fall in the population growth rate. (b) For the world, there is no limit to the fall in population if births remain below deaths. T4. Assuming that all countries in the world have entered the ‘modern growth stage’ (as defined in this chapter), present the analysis of the long-run steady state for the world population and its standards of living under the modern growth theory with exogenous labour-saving technical change. The modern growth stage opens with living standards that are already substantially above the subsistence level. In this stage, the population growth rate is now negatively related to the (further) improvements in the standard of living. Mortality has already declined considerably prior to this stage and does not
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Output units
(n + η)kE
σ f(kE)
kE1
kE*0
kE
Figure 14.T4
significantly fall further. But birth rates fall due to a decrease in the desired number of children per woman, which become feasible through an easily available low-cost birth control technology. Therefore, in the long run, the world population will decline. Hence, the labour force growth rate will be negative. The analysis of this stage can be based on the Solow model with the addition of the above assumptions of a negative labour force growth rate and positive labour-saving technical change. The following answer assumes that the sum of the labour growth rate and the rate of growth of labour efficiency, (n + η), is positive. The Solow model then implies that the improvements in the standard of living in the pre-SS stage will occur due to both falling population growth rates — which raise the growth rate of capital per worker — and technical change. In the SS, the standards of living will grow at the rate η. The relevant diagram is shown in Figure 14.T4. In this diagram, k1E illustrates the pre-SS position, with both increases in the capital–labour ratio and technical change. Living standards rise due to both these changes. k0E is the SS position with a constant KE /L ratio, with increases in living standards at the rate η due only to technical change. Basic growth model for this chapter: Y/L = 100(K/L) − 0.5(K/L)2 APC = MPC = 0.8 L = 0.04 T5. For the basic growth model of this chapter, calculate the marginal product of capital per worker (MPk). Is MPk constant in the steady state of the basic Solow model without technical change? Explain your answer. The production function is: y = 100K/L − 0.5(K/L)2 = 100k − 0.5k2 The marginal product of capital per worker (MPk) is: MPk = ∂y/∂k = 100 − k In the SS of the Solow model without technical change, MPk is constant because k is constant.
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T6. Using the basic growth model of this chapter, (i) What is the SS value of k? We know that σ = 1 − MPC = 1 − 0.8 = 0.2, that n = L = 0.04 and that f(k) = 100k − 0.5k2 . Substituting these values in the fundamental SS equation [σf(k) = nk] of the Solow model: 0.2(100k − 0.5k2 ) = 0.04k 20k − 0.1k2 = 0.04k 20 − 0.1k = 0.04 k = 199.6 (ii) What is the SS value of y? The SS value of y is given by: y = f(k) = 100k − 0.5k2 = 100(199.6) − 0.5(199.6)2 = 39.92 The level of saving per capita in the SS is given by: σf(k) = 0.2(39.92) = 7.984 (iii) What are the steady-state levels of (a) consumption per capita and (b) saving per capita? The level of consumption per capita can be found as follows: c = APC · f(k) = 0.8(39.92) = 31.936 The level of saving per capita in the SS is given by: σf(k) = 0.2(39.92) = 7.984 (iv) If the MPC changes to 0.7, recalculate your answers to the previous questions. We know that σ = 1 − MPC = 1 − 0.7 = 0.3, that n = L = 0.04 and that f(k) = 100k − 0.5k2 . Substituting these values in the fundamental SS equation of the Solow model: 0.3(100k − 0.5k2 ) = 0.04k 30k − 0.15k2 = 0.04k 30 − 0.15k = 0.04 k = 199.73 The SS value of y is given by: y = f(k) = 100k − 0.5k2 = 100(199.73) − 0.5(199.73)2 = 26.96 The level of saving per capita in the SS is given by: σf(k) = 0.3(26.96) = 8.09 Consumption is: c = APC · f(k) = 0.7(26.96) = 18.87 (v) What is the change in the SS growth rates of output and capital for the two values of MPC? There is no change in the SS growth rates of either output or capital. For both values of MPC, output and capital grow at the labour force growth rate, i.e., n = 0.04. Revised growth model A for this chapter, with labour-saving technical change: yE = 1,000kE − 2kE2 APC = MPC = 0.8 L = n = 0.02 and η = 0.02 [n is the growth rate of workers; η is the rate of growth of efficiency per worker.]
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T7. Using the revised growth model A of this chapter, (i) What are the SS growth rates of kE and k? (ii) What are the SS growth rates of yE and y? This question can be answered without using the information on the production function and the APC/MPC. Only the information of η = 0.02 is needed. The answer is as follows: yE = kE = 0 y = k = η = 0.02 Revised growth model B for this chapter, with labour-saving technical change: yE = 2,000kE − 4kE2 APC = MPC = 0.9 E = n + η n = 0.05, η = 0.015 T8. Using the revised model B of this chapter, (i) What is the SS value of kE ? To answer this question, we start with the fundamental SS equation for the Solow model with technical change: σyE = (n + η)kE We know n and η, and can compute σ = 1 − APC = 1 − 0.8 = 0.2. Therefore, 0.2(2,000kE − 4kE2 ) = (0.05 + 0.015)kE Dividing each side by kE : 400 − 0.8kE = 0.065 kE = 499.92 (ii) What is the SS value of yE ? The value of yE can be computed from the production function: yE = 2,000kE − 4kE2 = 2,000(499.92) − 4(499.92)2 = 159.97 (iii) What is the growth rate of consumption per capita in the steady state? Since consumption is a fixed fraction of output, it will grow at the same rate as output. At the SS, output growth will be equal to the growth rate of efficiency per worker η = 0.015. Therefore, the growth rate of consumption per capita will be 0.015. (iv) Suppose that because of a low birth rate, aging population with rising numbers of retirees and inadequate immigration levels, the labour force growth rate falls to −0.01, while η remains at 0.015. Recalculate your answers to (a) to (c). To answer this question, we start with the fundamental SS equation for the Solow model with technical change: σyE = (n + η)kE We know n and η, and can compute σ = 1 − APC = 1 − 0.8 = 0.2. Therefore, 0.2(2,000kE − 4kE2 ) = (0.01 + 0.015)kE Dividing each side by kE : 400 − 0.8kE = 0.025
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Macroeconomics: Answers to BOOK QS CH 14 Output units
(n0 + η 0)kE (n1 + η 0)kE
σ f(kE)
kE*0
kE*1
kE
Figure 14.T8
kE = 499.97 The value of yE can be computed from the production function: yE = 2,000kE − 4kE2 = 2,000(499.97) − 4(499.97)2 = 60.00 Note that the SS growth rate of consumption per capita will remain the same at 0.015. Use a diagram to illustrate the general nature of your results for this part of the question. In Figure 14.T8, the decrease in the labour force growth rate would be represented by a downward shift/rotation of the (flatter) nkE line, while an increase in rate of labour-saving technical change would be represented by an upward shift of the nkE line. Assuming that the rate of technical change remains constant at η0 , a decrease in the labour force growth rate from n0 to n1 rotates the (n + η)kE line down from (n0 + η0 )kE to (n1 + η0 )kE , and so the lower labour force growth rate results in a higher SS level of capital per worker, which would raise the SS output per worker.
