The determinant of the coefficient matrix of the homogeneous system is ( -<5: — a), which is negative. We therefore know immediately that the steady-state equilibrium is a saddle point.

Bv theorem 24.2, the solutions to the system of differential equations in (25.7) and (25.8) aie where r\ and r2 are the eigenvalues or roots of the coefficient matrix in equation (25.9), Ci and C2 are arbitrary constants of integration, and k and K are the steady-state values of the system, and serve as particular solutions in finding the complete solutions.

If A denotes the coefficient matrix in equation (25.9). its characteristic roots (eigenvalues) are given by the equation tr(A) 1 /-r-

where ir{A) denotes the trace of A (sum of the diagonal elements). The roots of equation (25.9) then are r|, rj = ±V5Z 4- a

The steady-state values of X and K arc found by setting X =s 0 and K = 0. Doing this and simplifying yields

Solving these for A and K give the steady-state values

Because the steady state is a saddle point, it can be reached only along die saddle path and only if the exogenously specified time horizon, T. is large enough to permit it to be reached.

This leaves only the values of the arbitrary constants of integration to he determined. As usual, they are determined using the boundary conditions K(0) -K,) and X(T I = 0. First, requiring the solution for K (/) to satisfy its initial condition gives

After simplifying, this gives

Next, requiring the solution for X(t) to satisfy its terminal condition gives 0 = C\er,T + Cj«?'-7 +X

from which we get an equation for C2 in terms of CV

Substituting this into the expression for C\ and simplifying gives the solution for C\\

2a(K{) - K) + (r2 - 8)ke"'-~lr{ -8)- (r2 - S)e<r<~r-)T

Figure 25.2 Solution path 1,(1) for investment when A'(, < K: solution path /2(i) for investment when K„ > K

Substituting this solution into the equation for C2 and simplifying gives the explicit solution for C2:

-lit(Ko - K)e{ri~r-,T -X(r, - 8)e.^-r n -8 - (r2- S)e(r'-r"-)r

This completes the solution.

The optimal path of investment is obtained using equation (25.6). If we denote the solution for k(t) in equation (25.10) as k*(t), then the solution for investment, denoted /*(/) is

This solution gives the path of investment that maximizes total profits over the planning horizon. Figure 25.2 shows two possible solution paths. When Ko < K. the solution is a path like l, (/) that starts high and declines monotonically to 0 at time 7. When Ka > K, the solution is a path of disinvestment like l2U) that stays negative from zero to T.

An Economic Interpretation of A and the Hamiltonian

We introduced k(/) as a sequence or path of Lagrange multipliers. It turns out that there is a natural economic interpretation of this co-state variable. Intuitively k(t) can be interpreted as the marginal (imputed) value or shadow price of the state variable x(t). This interpretation follows informally from the Lagrange multiplier analogy. But it also follows more formally from a result that is proved in the appendix to the chapter. There it is shown that /.(()) is the amount by which J' (the maximum value function) would increase if a(0) (the initial value of the stale variable) were to increase by a small amount. Therefore k(0) is the value of a marginal increase in the state variable at time i — 0 and therefore can be interpreted as the most we would he willing to pay (the shadow price) to acquire a hit more of it at time t = 0. By extension. ).(t) can be interpreted as the shadow price or imputed value of the stale variable at any time t.

In the investment problem just examined. A (?) gives the marginal (imputed) value or shadow price of the firm's capital stock at time /. Armed with this interpretation. the first-order condition (25.5) makes economic sense: it says that at each moment of time, the lirm should carry out the amount of investment (hat satisfies the following equality:


The left-hand side is the marginal cost of investment; the right-hand side is the marginal (imputed) value of capital and. as such, gives the marginal benefit of investment. Thus the first-order condition of the maximum principle leads to a very simple investment rule: invest up to the point that marginal cost equals marginal benefit.

The Hamillonian function loo can be given an economic interpretation. In general. H measures the instantaneous total economic contribution made by the control variable toward the integral objective function. In the context of the investment problem, // is the sum of total profits earned at a point in time and the accrual of capital that occurs at that point in time valued at its shadow price. Therefore H is the instantaneous total contribution made by the control variable to the integral of profits. J. It makes sense then to choose the control variable so as to maximize H at each point in lime. This, of course, is what the maximum principle requires.

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