P

We found the solution for ii(t), the current-valued costate variable, to be

To find die value of the constant of integration, Cwe need to use the boundary condition fx(T) = 0 fit* the constraint on x(T) is not binding] or x(T) = b [if the constraint on x(T) is binding]. To decide, try to solve for the unconstrained x'(T). Set e~pr/j,(T) = 0. Since T is finite, this amounts to setting fUT) = 0, which gives Ct = 0. However, inspection of equation (25.26) reveals that as C| —» 0, xIt) —» —oo for al) values of i. including I - T. This clearly violates the consu-aint that a (T) > b. Thus, we know that the relevant boundary condition is a(T) = b and that we should proceed to solve this as a fixed-endpoint problem. The solution, therefore, is the same as the fixed-endpoint version already solved, A related approach to deciding that the constraint is binding in this problem is to note that if C, -- 0. then p(i) ~ 0 for all t e (0. 7"). However, the first-order condition for r(z) indicates that as ;i(i) —«• 0. cii) —- oo. That is, Uie consumption rate becomes infinitely large as p.\t) approaches zero. This is clearly impossible, as it would not only drain tiie bank account but drive it to minus inlinity, as we determined above.

We conclude that in the absence of a bequest constraint in this model, the optimal consumption path is one that drives the bank account down to the minimum permitted level, by the end of the lifetime, T. The smallest value that b could possibly take in reality is b - 0. In that case the optimal consumption path involves consuming all of the capital in the bank account by time T ■

Optimal Depletion of an Exhaustible Resource

Imagine an individual stranded on a desert island where the only source of food is a lixed stock of an exhaustible resource. The resource is not perishable but also does not accumulate or reproduce. The individual is assumed to live from lime 0 (now) to T and is assumed to know this with certainty. The individual must choose a consumption path knowing that the stock of food is exhaustible. We assume the path chosen is the one that maximizes the discounted sum of utility. The maximization problem s where V[c(t)\ is instantaneous utility at i and p is the personal rate of time preference.

The differential equation shows that the stock of the resource, R(l), declines by the amount consumed. c(i). The resource stock starts out at size R{) and cannot decline below a size of zero. We now have an inequality-constrained endpoint problem.

The current-valued Hamiltonian function is

Assuming that U' > 0 and U" < 0 (positive and diminishing marginal utility) means the following condition holds on the optimal consumption path i max i max subject to R — —c

Implicitly this gives a solution lor c as a function of //. which we write as c = <(>(n)

In addition the costate variable must satisfy

since dH/dR = 0 in this model. The system of differential equations then is

The relevant boundary conditions are R{0) = Rq and either the transversal!ty condition p(T) = 0 or the endpoint condition R(T) = 0. The solution to the first differential equation is

where C| is a constant of integration. If we impose the condition /j(T) = 0, we obtain the solution C\ = 0, from which we conclude immediately that pit) = 0 for / e (U. T). Will this lead to the constraint R(T) > Obemg violated? The answer depends on the fonn of the utility function. If marginal utility falls to zero at some consumption level, say c. the solution to equation (25.27) with //(f) = 0isc(i) = c for all i e (0, T). Lifetime consumption is cT. If cT is less than Ro, then R(T) is clearly positive, so the resource constraint is nonbinding. In this case the resource is not scarce: more than enough of it is available to sustain the consumer's desired lifetime consumption, so it has no economic value. As a result its shadow price is zero.

On the other hand, if cT exceeds /?0, the resource constraint becomes binding, and ¡i(T) = 0 is the wrong boundary condition. R(T) — 0 becomes the correct boundary condition.

If the utility function does not have a satiation point (we typically assume that it doesn't), (hen marginal utility never becomes zero, no matter how high the consumption rate. This is the case for the utility functions we have examined so far in this chapter. For example, if U(c) - In c, then U'(c) = 1 /c, and this tends to zero only as c — oo. In this case the solution to equation (25.27) with p(t) = 0 is an infinitely high consumption rate. This would obviously drive the resource stock to negative infinity, thereby violating the constraint. Thus p(T) = 0 is the wrong boundary condition; R(T) = 0 is the correct boundary condition.

To complete the solution, we will assume that the resource is economically scarce, which means that we need to assume either that there is no finite satiation

t'

n

c*(0)

r*(T)

0 0

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