Macroeconomics ECO 332 Chapter 11 PDF

Macroeconomics, ECO 332 Chapter 11 Economic Growth Theory – Saving, capital accumulation and output The Solow growth theory Economic growth theory has evolved over the years, but efforts were intensified after the Second World War in the 1950s. The dominant growth theor...

Macroeconomics, ECO 332 Chapter 11 Economic Growth Theory – Saving, capital accumulation and output The Solow growth theory Economic growth theory has evolved over the years, but efforts were intensified after the Second World War in the 1950s. The dominant growth theory in the body of literature in economics is the Solow growth theory, often referred to as the neoclassical growth theory. The building blocks of the Solow growth model is derived from the circular flow model of the economy or from the Keynesian cross. We first, represent the relation between saving, capital accumulation and output with figure 11.1 2 Figure 11.1 Capital, output, saving/investment 3 1.1 Interactions between output and capital Figure 1 shows the relationship between capital and saving on one hand and capital and investment on the other. (a) Effects of capital on output 𝑌 𝐾 Recall from the last =𝐹 chapter: ,1 , we simplify this by 𝑁 𝑁 dropping, 1, which is a constant, anyway. 𝑌 𝐾 We have, =𝐹 𝑁 𝑁 Further assume: (1) Labour force, N is constant, (2) There is no technological change. 4 (a) Effects to capital on output cont’d Drawing on the assumptions: 𝒀𝒕 𝑲𝒕 =𝒇 1.1 𝑵 𝑵 (b) Effects of output on capital accumulation To analyse the effect of output on capital accumulation, we to find the relation between: Output and investment, Investment and savings. 5 (b) Effects of output on capital accumulation, cont’d We assume: The economy is closed, meaning , investment is equal to private saving (recall from the circular flow model of the economy or the Keynesian cross, I = S Savings is proportional to output: S = sY Combining the two equations above, we have, It = sYt 1.2a For the evolution of capital we have: K t+1 = 1 − δ K t +It 1.2b Now, substitute 11.2a into 11.2b , divide through by N and re-arrange to obtain, 𝐊 𝐭+𝟏 𝐊𝐭 𝐘 = 1−δ + 𝐬 𝐍𝐭 1.2 𝐍 𝐍 𝐊 𝐭+𝟏 𝐊𝐭 𝐘𝐭 𝐊𝐭 − =𝐬 −𝛅 1.2 𝐍 𝐍 𝐍 𝐍 Where, t stands for year t, and t +1, year t +1; 6 1.2 Implications of alternative saving rate Combine equations 1.1 and 1.2 to see the dynamics of capital and output over time. (a) Dynamics of capital and output Substitute 11.2 into 11.1, 𝐊 𝐭+𝟏 𝐊𝐭 𝐘𝐭 𝐊𝐭 − = 𝐬𝒇 −𝛅 1.3 𝐍 𝐍 𝐍 𝐍 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒄𝒂𝒑𝒊𝒕𝒂𝒍 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒚𝒆𝒂𝒓 𝒕 + 𝟏 𝒂𝒏𝒅 𝒚𝒆𝒂𝒓 𝒕 = 𝒊𝒏𝒗𝒆𝒔𝒕𝒎𝒆𝒏𝒕 𝒊𝒏 𝒚𝒆𝒂𝒓 𝒕 - 𝒅𝒆𝒑𝒓𝒆𝒄𝒊𝒂𝒕𝒊𝒐𝒏 𝒊𝒏 𝒚𝒆𝒂𝒓 𝒕 Equations 11.2 and 11.3 are critical in the Solow growth model because they contain all the information required to understand and analyse the dynamics of capital and output over time. 7  Figure 1.2 Capital and output dynamics  When capital and output are low, investment exceeds depreciation and capital Steady state increases.  When capital and output are high, investment is less than depreciation and capital decreases.  The graph is the easiest way to analyse the interpret the relation between capital and output 8 Conclusions ∗ ∗ When ൗ𝑁 reaches 𝐾 Τ𝑁 and 𝑌𝑡ൗ𝑁 also reaches 𝑌 Τ𝑁, capital and output have 𝐾𝑡 reached their long run equilibrium levels and as a result remain constant. ∗ 𝐘𝐭 𝐊𝐭 On the right of 𝐾 Τ𝑁: 𝐬𝒇 𝛅 𝐭, and 𝐾𝑡ൗ𝑁 increases 𝐍 𝐍 The steady state of the economy is the state at which capital per worker and output per worker are no longer changing but rather remain constant. Thus, definition, change in capital per worker is zero at the steady state. ∗ Set 1.3 to zero and we have the steady state Τ𝑁 being given by: 𝐾 𝒀∗ 𝑲∗ 𝐬𝒇 = 𝛅 11.4 𝐍 𝐍 9 Conclusions, cont’d ∗ Given the steady state of capital per worker 𝐾 as, Τ𝑁, the steady state 𝑌 ∗ output per worker, Τ𝑁 can also be represented by production function, 𝒀∗ 𝑲∗ =𝒇 1.5 𝑵 𝑵 10 (b) The saving rate and output Here we answer the question, how does savings rate affect output per worker? (i) Saving rate has no effect on long run growth rate of output per worker, which is equal to zero. Fig 3 shows the long run economy converges to constant level of output per worker at steady state. Think about the need to keep increasing 𝒔, till it reaches 𝟏, a feat which is impossible, in order to sustain increasing output per worker. (ii) Yet, saving rate determine output per worker in the long run. All things being equal, higher saving rate generates higher output per worker in the long run. 11  Fig. 3 Effects of different saving rates See, two countries with: the same production function; same depreciation rate, but different saving rates, 𝒔𝟎 , 𝒔𝟏 ; 𝒔𝟏 > 𝒔𝟎 In the long run, country with higher saving rate will reach higher levels of 𝐾/𝑁 and higher steady state of output per worker. 12 (b) The saving rate and output, cont’d (iii). An increase in the saving rate will generate higher growth per worker but not forever, or in sustained manner. from (i), increase in saving has no effect on long run growth; and (ii), increase in saving leads to increase in long run growth in output per worker. the increase in output per worker comes to an end when the economy reaches steady state. 13  Fig. 4 Effects of increase in the saving rate on output in an economy without technological progress An increase in the saving rate leads to a period of higher growth rate until output reaches higher new steady state level. 14  Fig. 5 Effects of increase in the saving rate on output per worker in an economy with technological progress An increase in the saving rate leads to a period of higher growth rate until output reaches higher new path. Output per worker grows at a constant rate: on both path 𝐴𝐴 and 𝐵𝐵, the growth rate is the same., ,i.e, both lines have same slope. Note, that we have a log scale in the 𝑌/𝑁 axis. 15 (c) The saving rate and consumption Gov’t can impact national saving rate by: Creating budget surplus, (𝑇 − 𝐺 > 0); Creating budget deficit, (𝑇 − 𝐺 < 0). The question then is, what saving rate is should government target? First, let see the relation between consumption and saving. Increases in saving do not necessarily produce higher level of consumption in the long run. 16 (c) The saving rate and consumption, cont‘d Let’s look at two extreme cases of saving rates. Zero saving : here capital per worker is also zero; output per worker is zero and consumption is also zero in the long run. Saving rate of 1: capital per worker will be very high, output per worker will be high as well, but consumption will be zero, why? There will be excess capital here, and all output per worker will be used to replace depreciation. Thus, there must be an optimal saving rate that maximises steady state level of consumption. And this must lie between zero and 1, savings rate. Savings rate below the optimal value means consumption falls initially, but increases in the long run. Increase in saving rate above the optimal value translates into a fall in consumption initially and in the long run. The level of capital associate with the increase in saving rate generate output that is not able to cover depreciation because it’s too small. 17  Fig. 6 The effect of saving rate on steady state consumption per worker The level of capital associated with the saving rate that is able to generate the highest level of steady state consumption is known as the golden-rule level of capital. Increases in capital beyond the golden rule of capital reduces steady state consumption. Beyond, 𝑆𝐺, the increase in capital is not adequate to generate output, large enough, to offset depreciation of capital. 18 1.3 Saving rate and consumption: Getting a sense of the magnitudes Assume a production function, 𝑌 = 𝐾. 𝑁 Divide both sides by 𝑁, 𝑌 𝐾. 𝐾 𝐾 𝐾 = = = 𝑁 𝑁 𝑁 𝑁 The production function linking capital and output may be given as: 𝐾𝑡 𝐾𝑡 𝑓 = 𝑁 𝑁 𝐾𝑡 If we replace, 𝑓 in equation 1.3, we obtain, 𝑁 𝐾𝑡+1 𝐾𝑡 𝐾𝑡 𝐾𝑡 − =𝑠 −𝛿 1.7 𝑁 𝑁 𝑁 𝑁 The equation describes the evolution or changes in capital over time. 19 (a) Effect of saving rate on steady state What is the effect of saving rate on output per worker? Recall, steady state LHS of eqn 1.7 is zero, this imply, 𝐾∗ 𝐾∗ 𝑠= = 𝛿 𝑁 𝑁 Square both sides; 𝐾∗ ∗ 2 𝑠 =𝛿 2 𝐾 𝑁 𝑁 Divide both sides by 𝐾Τ𝑁 and re-arrange to obtain, 𝐾∗ 𝑠 2 = 1.8 𝑁 𝛿 20 (a) Effect of saving rate on steady state, cont’d The conclusion here is that the steady state output per worker is given by, 𝑌∗ 𝐾∗ 𝑠 2 𝑠 = = = 𝑁 𝑁 𝛿 𝛿 A higher saving rate and a lower depreciation rate, produces higher output per worker. Steady state output per worker is equal to the ratio of saving rate to output per worker. What happens to output per worker if saving rate increases from 0.1 to 0.2 and depreciation rate remains at 0.1 21 (a) Dynamic effect of increase in the saving rate To pin down ideas about how changes in saving affect steady state level of output, let’s plug in numbers. First, from eqn 1.7, solve for capital per worker in year 0 and year 1, etc. Assume saving rate had always been equal to 10 percent; It increases from 10 percent to 20 percent, from year 0 to year 1. At year 0, capital stock is equal to steady state value associated with saving rate of 0.1 From eqn. 1.8: for year 0 and 1, we have; 𝐾0 0.1 2 = = 12 ; 𝑁 0.1 𝐾1 0.1 2 𝐾1 𝐾2 −1= - 0.1 1 , thus, = 1.1, use the same to solve for , and so on. 𝑁 1 𝑁 𝑁 22 If the exercise is done for a long period, the following graphs are obtained. The conclusion here is that, it takes a long time for output to adjust its higher level after an increase in the saving rate. Better still, an increase in the saving rate leads to a long period of growth. 23 Table 1. The saving rate and the steady state levels of capital, output and consumption 24