Golden Rule of Capital Accumulation - Essay Example

?Golden rule of capital accumulation and macroeconomic policy The long-run growth model or the Solow Growth Model was developed by Nobel laureate Robert Solow. This model discusses about the long run accumulation of capital stock that is consistent with the steady state rate of growth in the particular economy. This is the neo-classical model of economic growth. In 1946, Robert Solow and T. W. Swan independently developed this exogenous growth model. This steady state rate of growth is consistent with the economy’s natural rate of output. Policymakers use this model to understand why different countries grow at different rates and this is the main relevance of this model. But this steady state rate of growth can vary across economies (Solow, 1994, pp.45-46). In this context the Golden Rule of capital accumulation determines the optimal level of capital per capita that produces the optimal sustained level of per capita consumption in the economy (Mankiw, 2006, pp.110-115). This paper discusses this Golden Rule of capital accumulation and explains implications for macroeconomic policies in this context. Steady state level of capital and output per capita: The Solow Growth model determines the ‘steady state level of capital stock’ per capita and the steady state level of output per capita. But the Golden Rule of capital accumulation determines the maximum level of consumption per capita at the ‘steady state level of capital stock’ (Blanchard, 2006, p.230). This is called the sustainable level of consumption per capita. Here sustainable means that the present generation of the economy saves exactly that amount which is consistent with the replacement of the loss of capital stock that happened due to depreciation of that capital stock, no more and no less. In this sense this Golden Rule of capital stock provides the optimal level of consumption, savings and investment per capita at each period. Before deriving the Golden Rule capital stock let us first determine the steady state level of per capita capital and per capita output (Arnold, 2011, p.340). Assumptions and observations: Suppose that the aggregate production function is given by Y = F (K, L), where, “Y” =aggregate output level “F ( )” = aggregate function “K” =aggregate level of capital stock “L” = aggregate stock of labour (Solow, 1994, pp.45-54). Let, “n” be the constant and exogenous rate of growth of labour force. By dividing the aggregate production function by the stock of labour “L”, we get the per capita production function as, y = f (k, 1), where, “y” = per capita output “f ( )” =per capita function “k” = per capita capital stock “1” is the number. Or this per capita production function can be written as y = f (k). The production function in this economy is assumed to describe the assumption of diminishing marginal productivity, i.e. rate of change in output per capita declines with the increase level of capital stock per capita. That is why the per capita production function is upward sloping and concave. The production function may exhibit constant returns to scale, i.e. one unit increase in the per capita capital raises output per capita by one unit (Baumol, 1986, pp.1072-1101). “?” is assumed to be the constant rate at which capital stock depreciates in each period. Hence, the total depreciation of capital per capita is: (?+n)*k. Assuming “s” as the constant rate of saving per capita, the total level of savings in the economy will be: s*y = s*f (k) As savings rate equals investment, Economy’s investment is given as s*f (k) (Jones, 2002, pp.97-104). “dk/dt” measures the rate of change of capital stock per capita and is computed as dk/dt = s*y - (?+n)*k, where “t” = time element (for simplicity writing the “t” notation is avoided in each function). Hence, the ‘steady state level of capital stock’ is achieved for that level of capital stock per capita where the change in capital stock is zero, i.e. , where dk/dt = 0. The steady state capital stock is denoted by k* and the steady state level of output as y* (McEachern, 2009, pp.165-171). The method of transition of the capital stock per capita to the steady state level can be described by stable equilibrium analysis. If the current per capita capital stock is at k1, then level of per capita savings or investment in capital stock in the economy is higher than the amount depreciated. Hence, capital stock per capita will continue to rise until it reaches the steady state level k*. Similarly if the current per capita capital stock in the economy is at k2, then it will fall in the next period , the amount of capital per capita depreciated being higher than the amount replaced by investment (Barro, 2008, pp.68-70). These steady state levels of capital stock per capita and output per capita are shown below: Figure 1: Steady state level of capital and output per capita Golden Rule capital stock: The Golden Rule level of capital stock is achieved by maximizing the level of consumption per capita on the steady state growth path of the economy. Here consumption is nothing but the difference between the steady state level of output per capita and steady state level of savings per capita (Edmund Phelps’s Contributions to Macroeconomics, 2006, pp.10-19). “c” is the per capita consumption in the economy. Hence, c = C/L, where, “C” = aggregate consumption level in the economy “c” being a function of k*, the condition for achieving the Golden Rule capital stock is given by (Gartner, 2003, p.176): Max c (k*) = f (k*) – s*f (k*) Hence, solving the above mentioned problem we get the condition for the Golden rule capital accumulation as: df/dk* = ?+n, df/dk* being the marginal productivity of steady state capital stock per capita. This condition describes that at the golden rule capital stock level, the marginal product of steady state capital stock equals the sum of depreciation rate and the rate of growth of labour (Branson, 1979, pp.53-58). this condition can also be written as: df/dk* - ? = n stating that net marginal product of steady state capital stock (equivalent to the gross marginal rate of product of steady state capital stock net of depreciation) should be equal to the rate of growth of labour force (Chamberlin and Yueh, 2006, pp.553-558). By solving the condition for Golden Rule we get the Golden Rule level of capital accumulation “cGold*”. The Golden Rule level of per capita savings and per capita investment are sGold* and iGold* respectively. The transition process towards equilibrium can be explained by the stable equilibrium analysis. If the level of ‘steady state per capita capital’ is lower than the Golden Rule level (for example, k1*), then steady state consumption can be raised adding one more unit of ‘steady state per capita capital’. Hence, steady state consumption will continue to rise till Golden Rule level of per capita capital stock is reached. Similarly the steady state per capita consumption will fall as long ‘steady state level of per capita capital’ is higher than the golden Rule level (such as at k2*). Final equilibrium will restore at kGold*, with unchanged consumption per capita (Burda and Wyplosz, 2009, pp.71-72). The figure below shows the Golden Rule of capital accumulation (Edmund Phelps’s Contributions to Macroeconomics, 2006, pp.10-19). Figure 2: Golden Rule of capital accumulation Macroeconomic policy implications: There are important macroeconomic policy implications for investment savings under Golden Rule of capital accumulation as discussed below (Thirlwall, 2002, p.69): The major macroeconomic policy implication of Golden Rule is that it explains why different countries have different levels of income and capital stock and are growing at different rates. To achieve the steady state levels of per capita capital and per capita output, the macroeconomic policymakers need to implement policies for increments in rate of savings and investment. For instance if the economy is below the level of Golden Rule level of capital stock then the economy needs to increase it rate of savings thereby raising economy’ investment. One unit increase in the rate of savings will raise the level of per capita capital employed in the production process by one unit and hence lead to unitary increase the per capital output produced in the economy. Thus level of per capita consumption will increase by c unit. Again as consumption rises, demand for goods and services increase and employment of per capita capital rises too. Thus per capita output increases. Hence, per capita consumption and demand rises again. the process continues till the steady state levels of per capita capital stock and per capita output are achieved at which economy’s per capita consumption is achieved (Carlin and Soskice, 2006, pp.477-481). This is the Golden Rule level of capital accumulation (see figure 3). If the objective of macroeconomic policymakers is to increase the rate of growth of the economy under consideration, then the suitable macroeconomic policy can be increasing the economy’s rate of savings to increase the level of per capita capital stock through increase in economy’s investment. Many developed nations like USA, Japan have followed this method for higher rate of growth. When World War II destroyed Japanese economy, the country saved and invested major portions of national income to increase its capital stock. This phenomenon is known as “economic miracle” (Gnos and Rochon, 2011, p.279). (Figure A) (Figure B) Figure 3: Implication of change in savings rate on capital per capita and output per capita (Figure A) and on Golden Rule (Figure B) Conclusion: The ‘steady state level of capital stock’ and the Golden Rule of capital accumulation are important for any economy to determine present status growth rate and also help policymakers to evaluate and implement the necessary policies in order to achieve the desired rate of growth. But the most important implication of these concepts is related to the policy of catching up the developed or high-rate-of –growth experienced countries. These policies are related to the policies discussed above. References: 1. Arnold, R. A. (2011). Macroeconomics, USA: Cengage 2. Barro, R. (2008). Macroeconomics: A Modern Approach, USA: Cengage 3. Baumol, W.J. (l986) Productivity Growth, Convergence and Welfare: What the Long Run Data Show, American Economic Review, Vol. 76, No. 5, pp. 1072-1101, available at: http://www.jstor.org/pss/1816469 (accessed on January 2, 2012) 4. Blanchard, O. (2006). Macroeconomics, USA: Pearson 5. Branson, W. (1979). Macroeconomic Theory and Policy, London: Routledge 6. Burda, M. and Wyplosz, C. (2009). Macroeconomics: A European Text, Oxford: Oxford University Press 7. Carlin, W. and Soskice, D. (2006). Macroeconomics, Imperfections, Institutions and Policy, Oxford: Oxford University Press 8. Chamberlin, G. and Yueh, L. (2006). Macroeconomics, USA: Cengage 9. Edmund Phelps’s Contributions to Macroeconomics (2006). Kungl. Vetenskapsakademien: The Royal Swedish Academy of Science, available at: http://people.su.se/~calmf/NobelPhelps2006Adv.pdf (accessed on January 2, 2012) 10. Gartner, M. (2003). Macroeconomics, USA: Pearson 11. Gnos, G and Rochon, L-P. (2011), Credit, Money and Macroeconomic Policy: A Post-Keynesian Approach, UK: Edward Elgar Publishing 12. Jones, C. I. (2002). Introduction to Economic Growth, New York and London: W.W. Norton and Co. 13. Mankiw, N. G. (2006). Macroeconomics, USA: W H Freeman 14. McEachern, W. A. (2009). Macroeconomics: A contemporary Introduction, USA: Cengage 15. Solow, R. (l994) Perspectives on Growth Theory.Journal of Economic Perspectives. The Journal of Economic Perspectives, Vol. 8, No. 1, pp. 45-54, available at: http://teaching.fec.anu.edu.au/econ2102/2007/references/solowjep94.pdf (accessed on January 2, 2012) 16. Thirlwall, A. P. (2002). The Nature of Economic Growth: An Alternative Framework for Understanding the Performance of Nations, UK: Edward Elgar Read More