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The Ramsey Model of growth

Frank Ramsey  (Philosopher and Mathematician ) derived a theory about inter-temporal consumption and optimal saving function using the standard utilitarian philosophy of Jeremy Bentham. The basic point of Ramsey's Model is that it is a prescriptive theory and not merely predictive. Society has to choose optimal economic growth. Society as being represented by 'representative agent' or a central planning agency. This representative agent or central planner maximizes an objective function. 
In optimal growth of Ramsey type the utility function most often used is 
U = U(C/L)
or U = U(c) where c = C/L
above equation means that only consumption is used as a proxy for all welfare inducing activity .


ramsey model


Let the time period be 0,1,2,......T(starting period is denoted by 0) and there is consumption stream in this period (C₀.C₁C₂,.....Cᴛ). The optimization problem is to choose that consumption sequence that maximizes utility.
Let U(0), U(1), U(2),...U(T) be the value of utility function at times 0,1,2......t. Thus the value of the objective function at time t is U(t). The idea of optimization is to maximize the value of objective  function over the entire path, which is
T
∑ U(t)
𝘵₌0
Since social welfare function or utility depends on the consumption per person at each time period, and optimization maximizes the value of objective function over the entire consumption path or sequence. thus maximization

        T                           
J =  Σ U [C(t)]
      𝘵₌0
1. here J is the value of the sum of utility function, and utility in each period is independent of other periods.Total utility is the sum of utility in all periods.
2. Here time just highlight that consumption depends on time  
3. If we use continuous time then objective function will be
         T                          
J = ഽ U [c(t)] dt
       0
4. The term J is Functional. this means even though utility depends on (c) and (c) depends on (t) it does not mean that U becomes indirect function of t. U is dependent on whole path or the whole function c. 
The basic idea of Ramsey model is presented above. 
                 
                                         Mathematically

He thought of an ideal level of utility, called bliss point and saw how much actual utility fell short of it. 
Let B denote ideal utility, the bliss point, which is same in all time periods. Let U(cₜ) be the actual utility from per worker consumption level c. Then, define a function B-U(cₜ) for net utility in period t. .Thus for the total path or the whole path, the functional to be optimised becomes
      T
J =∑ B-U(c)
     𝘵₌0
 Here B-U(cₜ)  represents the net utility obtained from t-0 consuming ct at time t. Of course, Ramsey used this optimization subject to the constraint of growth of capital stock in the economy, The main issue is whether the time horizon is to be taken to be up to some finite time period like T or should it extend to infinity. We come to that now. Now we discuss certain issues that present themselves in the Ramsey model. We discuss the issue of time horizon, and then we discuss a very important issue of discounting.  

If we take time to be finite and we find that some capital stock is left over at the terminal time period T, then optimum has not been reached and
        T
J =  Σ U [C(t)]
       𝘵₌0
would not have been optimized. The solution to this, and to make the problem mathematically tractable is to specify a terminal condition that some capital is retained at time T, and to build this condition into the optimization problem itself. To get around this problem and to make the solution  mathematically less difficult the time horizon is often taken to be till infinity.

The discounting problem can be posed as follows. Given that people do not take the future as seriously as the present, in the sense that from the point of view of the Economic Growth present, individuals tend to underestimate the future problems, should future values discounted? Ramsey held this to be morally and ethically indefensible. He insisted I that future consumption and welfare ought not to, be given less weight than current I consumption and welfare. Harrod, too, was against discounting the future. Some have I argued that since social welfare is taken to be proxied by the utility of the representative agent, and individuals are taken to be myopic and do discount the future, social welfare ought to be discounted. However, it can be counter-argued that I the future need not involve the same representative agent, but actually concerns 1 future generations, who are not present at the time when the current generation I undertakes its inter-temporal optimization exercise. The idea is that social welfare should not depend only on the current members of society, but of all members of i I society who will live in this society in future. This again has been contested on the ground that all democratic societies make policies for and consider votes of, current members of the society. 




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