ike the Solow–Swan model , the Ramsey–Cass–Koopmans model starts with an aggregate production function that satisfies the Inada conditions , of Cobb– Douglas type, , with factors capital , labour , and labour- augmenting technology . The amount of labour is equal to the population in the economy, and grows at a constant rate . Likewise, the level of technology grows at a constant rate . The first key equation of the Ramsey– Cass–Koopmans model is the law of motion for capital accumulation: where k is capital intensity (capital per worker), is change in capital per worker over time ( ), c is consumption per worker, f(k) is output per worker, and is the depreciation rate of capital. Under the simplifying assumption that there neither population growth nor an increase in technology level, this equation states that investment , or increase in capital per worker is that part of output which is not consumed, minus the rate of depreciation of capital. Investment is, therefore, the same as savings. It also yields a potentially optimal steady-state of the growth model, in which , i.e. no (further) change in capital intensity. Now, an has to determine the steady-state which maximizes consumption , and yields an optimal savings rate . This is the “golden rule ” optimality condition proposed by Edmund Phelps in 1961. [4] where I is the level of investment, Y is level of income and s is the savings rate, or the proportion of income that is saved. The second equation concerns the saving behavior of households and is less intuitive. If households are maximizing their consumption intertemporally, at each point in time they equate the marginal benefit of consumption today with that of consumption in the future, or equivalently, the marginal benefit of consumption in the future with its marginal cost. Because this is an intertemporal problem this means an equalization of rates rather than levels. There are two reasons why households prefer to consume now rather than in the future. First, they discount future consumption. Second, because the utility function is concave, households prefer a smooth consumption path. An increasing or a decreasing consumption path lowers the utility of consumption in the future. Hence the following relationship characterizes the optimal relationship between the various rates: rate of return on savings = rate at which consumption is discounted − percent change in marginal utility times the growth of consumption. Mathematically: A class of utility functions which are consistent with a steady state of this model are the isoelastic or constant relative risk aversion (CRRA) utility functions , given by: In this case we have: Then solving the above dynamic equation for consumption growth we get: which is the second key dynamic equation of the model and is usually called the Euler equation . With a neoclassical production function with constant returns to scale, the interest rate, r, will equal the marginal product of capital per worker. One particular case is given by the Cobb–Douglas production function which implies that the gross interest rate is hence the net interest rate r Setting and equal to zero we can find the steady state of this model.
Posted on: Fri, 21 Mar 2014 06:21:28 +0000
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