The Solow Model and Ramsey Model of Capital Accumulation - Math Problem Example

Compare and contrast the Solow model and Ramsey model of capital accumulation Introduction The purpose of this research is to compare and contrast two economic growth models – Solow model and the Ramsey model of capital accumulation – to understand whether economic growth influences income and wealth distribution. In all the economic growth research, experts take the Solow model as the starting point for every analysis that they conduct on economic growth (Barro). However, the Solow model has its own limitation and does not offer a method of endogenous savings. On the other hand, the optimal growth theory of Ramsey from the late 1960s has influenced the consumers’ behaviour modeling (King 128). However, such an approach is generally linked with superior dimensional dynamic systems. Therefore, using this approach most often makes the analysis intractable, even when the economic growth problem is a simple one. In the neoclassical growth theory, there are primarily two major modeling frameworks, namely the Solow model and the Ramsely growth model of capital accumulation. The most important distinguishing factor between these two models is the difference related to behaviour of consumers (Cass 235). For instance, the Solow model is known to establish credible consumption function with the support of some empirical data. However, the Ramsey model in its analysis assumes that the economy is inhabited by just one immortal representative household. This household optimally uses the consumption plan over a span of infinite time in an environment that is more institutional in nature (King 909). Further, the household transforms its wishes into actual allocation of resources at the given point of time. Also, the Ramsey model endogenously calculates consumption and saving, whereas, the Solow model believes that the representatives keep aside some significant and predictable portion of its output on a regular basis for the purpose of capital accumulation (Swan 345). Nonetheless, most of the other aspects of the Ramsey model is similar to that of the Solow model. For instance, the variables, such as Y(t), N(t), K(t), r(t), k(t), and w(t), used in the Ramsey model have similar connotations just as the Solow model (Uzawa 18). The Ramsey growth model is different from the Solow model in one very essential parameter, that is, the Ramsey model clearly models the consumer behaviour and endogenises saving, which is missing from the Solow model. Therefore, in the Ramsey model the saving rate in the general sense would not remain constant and also there is no uniformity in the confluence of the economy to its steady state. This is however, not the case with Solow model (Lucas 10-25). Another important feature of the Ramsey model’s endogenous saving rate is that the model is Pareto optimal. This means that the saving rate is at the golden rule stage. It is interesting to note that the outcome is Pareto optimal and not merely an inference of the endogenous savings rate. This in turn means that the Pareto optimality also needs situations that would help to hold the premises for the first welfare theorem. However, in the Solow model, the savings outcomes might result in becoming dynamically inefficient (Rebelo 551). Originally when the Ramsey model was established, it was set as a model to find solutions for a central planner's problem to maximise the levels of consumption through consecutive generations. It was in the later stages that the model was adopted by successive researchers to describe the decentralised dynamic economy (Mulligan 775). In order to understand these above given comparisons, it is important to comprehend the models individually. In the next section of the research paper, I would attempt to understand both the models separately and would bring out the concepts unique to each model. Solow Growth Model Basics The starting point of this growth model is the CSR or constant returns to scale production function, i.e. Y (t) = F (K(t), A(t)L(t)) (Solow 65). In this equation, the terms used are defined as: Y (t) equivalent to output K(t) equivalent to capital stock A(t) equivalent to knowledge L(t) equivalent to labour The technological progress is said increase the labour as soon as A and L are entered multiplicatively in the above equation. This indicates that the K/Y ratio would remain constant in the steady state. Therefore, it would be easier to utilise the CRS property in order to rewrite the production functionality in its intensive form through the multiplication of all terms by 1/AL (Solow 68). In this above equation, y = Y /AL and k = K/AL. These are the output and capital per unit of effective labour. Dynamics This model is positioned in a continuous time period. It is assumed that Labor (L) and knowledge (A) are exogenous and would increase at exponential rates as per the below equations (Solow 73): According to this assumption, a fraction s of the result is dedicated to savings, which in turn gets devoted to investment and a fraction (1−s) of the result is allocated for consumption (Solow 77). Thus, the s in the below equation is an exogenous saving rate. Further, the capital is developed as per: The same equation when written in its intensive form produces the following result: Further, it is easy to find the growth of y(t) = k(t)α and c(t) = (1 − s)y(t) from the above equation (Solow 81). Steady State The capital per unit of effective labour remains constant in the steady state (i.e., ˙k(t) = 0). The last equation means that the total savings which is also calculated on the basis of per unit of effective labour sk(t)α must be equal to or break-even investment (n + g + δ)k(t). Commonly, this steady state level of capital is represented as k∗. Further, if k(t) > k∗, savings would not be adequate enough to substitute the capital that was lost to depreciation, technological progress and labour growth. This happens as capital reveals decreasing marginal returns. This would also result in the fall of capital stock (˙k (t) < 0) and return to k∗. Further, the opposite also holds true in case k(t) < k∗. This would mean that system can be considered stable (Solow 82-85). Balanced Growth Path In spite of having any preliminary value of k(0), the economy would at last reach the steady-state level k∗. Also, the actual capital stock and not per unit of effective labour would grow at rate n+g, as k∗ = K∗/AL, which means that the economy is in a steady state. Further, as AL also increases at rate n+g, it means that Y would increase at rate n + g. Output per person Y /L is often been used as a measure of standards of living around various countries and across time. Along this balanced growth path, Y /L would increase at rate g (Solow 87). Major findings Some of the major findings of the Solow model (Solow 88-92) are: For any preliminary k(0), the economy would join with a balanced growth path wherein Y /L would increase at the exogenous rate of technological progress, g. This model also has a unique saving rate (sg). This helps in maximising consumption per worker, C/L. This is sometimes called the golden-rule saving rate. Further, this is specified by the condition f 0(k∗) = n + g + δ. This model also propagates conditional convergence. This means that even poor countries with smaller capital stocks can grow fast and would finally match up the richer nations. This would happen once these poor countries control the difference in labour growth rates, depreciation and savings rates. However, this model also has some shortcomings as well (Solow 93-95): The steady-state growth in Y /L is totally exogenous. The micro fundamentals of firm and household decisions are based on assumptions. This limitation is addressed by the Ramsey model. This model is also not able to elucidate on the huge disparities in living standards across time or across various countries. The model needs idealistically large differentiation, which can be either across time or countries, in technology and/or capital labour ratios. Ramsey Growth Model The Ramsey model takes the Solow model further by incorporating the explicitly optimal behaviour by households and firms (Ramsey 543). Basics The model is based on the following assumptions: Presence of huge number of similar households with every member giving one unit of labour The preliminary amount of labour is positioned at unity (L(0) = 1) and there isn’t any growth of labour (n = 0) The firms are owned by the households Every firm hires labour and rents out capital in the competitive input markets.Also they sell their output in the competitive output market These firms have right to the function of CRS production wherein A(t) grows at rate g and A(0) = 1 The capital does not have any depreciation (i.e., δ = 0) Household Behaviour It has been assumed that households are infinitely lived and the discounted stream of future utility is maximised through (Ramsey 545): In the above equation, C(t) is the control variable at every point of time. It is assumed that the instantaneous utility function u[C(t)] is in the constant elasticity of substitution (CES) class (Ramsey 546): In this equation, θ = −Cu00/u0 > 0 is the relative risk aversion coefficient. Bigger θs indicates less readiness to replace consumption inter-temporally and more curvature in the utility function. Here σ = 1/θ is generally known as the inter-temporal elasticity of substitution. Further, because θ → 0, the utility function turns into linear in consumption and σ → ∞ (Ramsey 550). The constraints of the households are provided by three factors, the first one being the flow budget constraint In this, the sole asset is a(t), the wage rate is w(t) is and the asset rate of return is r(t). The second factor is the no Ponzi-game condition and in the third condition a(0) is given. However, in order to solve this continuous time dynamic issue, one needs to use calculus of variations. In order to do so, start with the present value Hamiltonian (Ramsey 551-53) In this equation, the costate variable λ(t) is the current value shadow cost of income. Further, for maximisation, the first-order conditions would be: After combining the above two, we would get the following, which is also known as the Euler equation for consumption: Firm Behaviour As compared to household behaviour, firm behavior is simpler to understand. Most firms generally choose capital and labour to increase profits per period (Ramsey 553-56) The same equation when written in its intensive form looks like the following: As firms take w(t) and r(t) as given, they will give capital to the point wherein their marginal product is equal to the rental rate or This would also provide zero economic profits incase the labour is being paid its marginal product or Equilibrium Dynamics and Welfare In this kind of an economy, the equilibrium dynamics is explained through the equation of the capital accumulation (Ramsey 557): And Together with the appropriate substitutions and letting C(t) = egtc(t) and a(t) = egtk(t), the equilibrium for this kind of economy reduces to the below mentioned equations, wherein with the k(0) and transversality condition are given. Major findings The model produces steady-state (˙k = ˙ c = 0) growth due to the unique (c, k) combination. There is a unique c(0) for a given k(0). This would further lead to a non-divergent path towards a steady state. This path is called the saddle path, whereas the general property is called the saddle-path stability. This model endorses the first welfare theorem of economics according to which the competitive equilibrium is Pareto optimal. As the markets in Ramsey model are competitive therefore, the agents have to prove themselves by downgrading the other agents. In the Ramsey model, which is also known as the modified golden-rule level, the steady-state level of consumption per worker is less than the Solow model’s golden-rule level. This is due to the fact that permanently lower level of future consumption is gladly swapped for higher level of consumption by impatient optimising agents. Conclusion Through this study I have attempted to compare and contrast two economic growth models – Solow model and the Ramsey model of capital accumulation. My research also establishes the same basis that all the economic growth research starts from the Solow model and even the Ramsey model takes the Solow model further by incorporating the explicitly optimal behaviour by households and firms. However, the biggest distinguishing factor between these two models is the difference related to consumers’ behaviour. In the Ramsey model the saving rate in the general sense would not remain constant and also there is no uniformity in the confluence of the economy to its steady state. This is however, not the case with Solow model. This model propagates conditional convergence and assumes that even poor countries with smaller capital stocks can grow fast and would finally match up the richer nations. Further, the Solow model is known to be based on credible consumption function and empirical data. However, the Ramsey model assumes that the economy is inhabited by just one immortal representative household, which optimally uses the consumption plan over a span of infinite time in an institutional environment. Therefore, in the Solow model the micro fundamentals of firm and household decisions are based on assumptions. This limitation is addressed by the Ramsey model. Also, the Ramsey model endogenously calculates consumption and saving, whereas, the Solow model believes that the representatives keep aside some significant and predictable portion of its output on a regular basis for the purpose of capital accumulation. The Solow model has a unique saving rate (sg). This helps in maximising consumption per worker, C/L. This is sometimes called the golden-rule saving rate. However, the Ramsey model is known as the modified golden-rule level, and the steady-state level of consumption per worker is less than the Solow model’s golden-rule level. This is due to the fact that permanently lower level of future consumption is gladly swapped for higher level of consumption by impatient optimising agents. References Barro, Robert J. and Xavier Sala-i-Martin. Economic Growth. The MIT Press. 2nd Edition. Cass, D. “Optimum growth in an aggregative model of capital accumulation.” Review of Economic Studies 1965: 235-240. King, R.G. and S.T. Rebelo. “Public policy and economic growth: developing neo-classical implications.” Journal of Political Economy 1990: 126-150. King, R.G. and S.T. Rebelo. “Transitional dynamics and economic growth in the neo-classical growth model.” American Economic Review 1993: 908-931. Lucas, R.E. “On the mechanics of economic development.” Journal of Monetary Economics 1988: 3-42. Mulligan, C.B. and X. Sala-i-Martin. “Transitional dynamics in two sector models of endogenous growth.” Quarterly Journal of Economics 1993:737-773. Ramsey, F. “A mathematical theory of savings.” Economic Journal 1928: 543-559. Rebelo, S.T. “Long-run policy analysis and long-run growth.” Journal of Political Economy 1991: 500-521. Solow, R.M. “A contribution to the theory of economic growth.” Quarterly Journal of Economics 1956: 65-94. Swan, T.W. “Economic growth and capital accumulation.” Economic Record, 1956: 334-361. Uzawa, H. “Optimal technical change in an aggregative model of economic growth.” International Economic Review 1965:18. Read More