September 21, 2020, Christopher D. Carroll HamiltonianVSDiscrete
This handout solves a continuous-time version of the Ramsey/Cass-Koopmans (RCK) model using the Hamiltonian method, and shows the relationship between that method and the discrete-time approach.
The problem is to choose a path of consumption per capita from the present moment (arbitrarily called time 0) into the inﬁnite future, , that solves the problem
where is the time preference rate, is the population growth rate, and is the depreciation rate. (In continuous time, we think of all variables as implicitly being a function of time, but it is cumbersome to write, e.g., everywhere, so the time argument is omitted; we are also thinking of the initial value of capital at date 0 as being a ‘given’ in the problem, so that for some speciﬁc value of ).
To emphasize the similarity between the continuous-time and the discrete-time solutions where we have typically used the roman to denote value, for the continuous-time problem we deﬁne ‘curly’ value as a function of the initial level of capital as .
The current-value (discounted) Hamiltonian is
where is the state variable, is the control variable, and is the costate variable.
is the continuous-time equivalent of a Lagrange multiplier, so its value should be equivalent to the value of relaxing the corresponding constraint by an inﬁnitesimal amount. But the constraint in question is the capital-accumulation constraint. Thus should be equal to the value of having a tiny bit more capital, . In other words, you can think of .
The ﬁrst necessary Hamiltonian condition for optimality is
Note the similarity between (4) and the result we usually obtain by using the Envelope theorem in the discrete-time problem,
Thus, you can use the intuition you (should have) developed by now about why the marginal utility of consumption should be equal to the marginal value of extra resources to understand this Hamiltonian optimality condition.
The second necessary condition is
which expresses the growth rate of at an annual rate (because the interest rate and time preference rate are measured at an annual rate).
To interpret this in terms of our discrete-time model, begin with the condition
The ﬁnal necessary condition is just that the accumulation equation for capital is satisﬁed,
This is the continuous-time equivalent of what we have previously called the Dynamic Budget Constraint.
Up to now in this course we haven’t thought very much about what the time period is. Generally, we have expressed things in terms of yearly rates, so that for example we might choose and to represent an interest rate of 4 percent and a discount rate of 4 percent.
One of the attractive features of the time-consistent model we have been using is that it generates self-similar behavior as the time interval is changed. Thus if we wanted to solve a quarterly version of the model we would choose and and it would imply consumption of almost exactly 1/4 of the amount implied by the annual model, so that four quarters of such behavior would aggregate to the prediction of the annual model.
To put this in the most general form, suppose and correspond to ‘annual rate’ values and we want to divide the year into periods. Then the appropriate interest rate and discount factor on a per-period basis would be and . Thus the discrete-time equation could be rewritten
where the time interval is now th of a year (e.g. if =52, we’re talking weekly, so that period is one week after period ). Now we can use our old friend, the fact that , to note that this is approximately
We deﬁned the interest rate and time preference rate on an annual basis, but the time interval between and is only th of a year. Thus expresses the speed of change in at an annual rate.
Now, note that since the eﬀective interest rate in this model is , equation (10) is basically the same as (6) since and . Hence, the second optimality condition in the Hamiltonian optimization method is basically equivalent to the condition from the discrete-time optimization method!
The ﬁnal required condition (the transversality constraint) is
The translation of this into the discrete-time model is
Consider the simple model with a constant gross interest rate and CRRA utility. In that model, recall that . Thus considered from time zero (12) becomes
What this says is that you cannot behave in such a way that you expect to grow faster than the interest rate forever.1 This is the inﬁnite-horizon version of the intertemporal budget constraint. Among the inﬁnite number of time paths of and that will satisfy the ﬁrst order conditions above, only one will also satisfy this transversality constraint - because all the others imply a violation of the intertemporal budget constraint.
Now diﬀerentiate (??) with respect to time
and substitute this into equation (6) to get
using the fact derived earlier that for a CRRA utility function „ this becomes