Sections on the following topics have been rewritten in a clearer and more accessible style: the straight line, some applications of translations, exponential functions, hyperbolic functions, optimisation of functions of one variable, and the inverse matrix.
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More magazines by this user. Close Flag as Inappropriate. You have already flagged this document. Thank you, for helping us keep this platform clean. With that modification the answer in the book is quite correct. Actually, the given differential equation has no solution defined on the entire real line. The lower semicircle shown in Fig. The figure in answer in the book is misleading since it shows a full circle. Faster growth per capita is to be expected because foreign aid contributes positively.
So foreign aid must grow faster than the population. It follows from 5. Formula 5. According to 5. According to Problem 5.
In equilibrium, capital per worker increases as the savings rate increases, and decreases as the growth rate of the work force increases. The necessary first-order condition for this to be maximized w. If condition 3 in Theorem 5. Note that the constants A and B here are different from 0.
Since u1 and u2 are not proportional, Theorem 6. The general solu- tion of the corresponding homogeneous differential equation is therefore Case I in Theorem 6. The general solution of the homogeneous equation is therefore Case II in Theorem 6. The constants A and B are determined by the initial conditions. To find the solution with the given initial conditions we must determine A and B. Note that Q did not appear in the last equation. That is because Qt is a solution of the homogeneous equation.
The equation is not stable for any values of the constants. We can also see the instability from the criterion in 6. The system has a single equilibrium point, namely 0, 0. It seems clear from the phase diagram that this is not a stable equilibrium. Indeed, the equations show that any solution through a point on the y-axis below the origin will move straight downwards, away from 0, 0. The matrix A in Theorem 6. These conclusions are confirmed by the solution to part b.
See Section B. We must therefore increase the degree of the polynomial factor and try with a quadratic polynomial instead. Differentiating a w. Differentiating once more w. These are the same as the solutions of the characteristic equation of the differential equation e.
This is no coincidence. See the remark above Theorem 6. Therefore, f1 x g2 y the function H x, y is constant along each solution curve for the system. We have proved that L is a strong Liapunov function for the system, so x0 , y0 is locally asymptotically stable.
It looks as if x has a special role here, but that is an illusion. If we use the recipe with y instead of x as the independent variable in equations 7. Actually, these solutions are just two ways of writing the same thing.
Hence, by Theorem 8. Thus the graph of x is a straight line, namely the straight line through the points t0 , x0 and t1 , x1. Of course, this is precisely what we would expect: the shortest curve between two points is the straight line segment between them. Hence, Theorem 8. Thus planting will take place at the constant rate of hectares per year. The value of A here is twice the value of A given in the book. For the rest see the answer in the book. To check the maximum condition 9.
This solution is valid if the required total production B is large relative to the time period T available, and the storage cost a is sufficiently small relative to the unit production cost b. See Kamien and Schwartz for further economic interpretations.
Then we guess that production is postponed in an initial time period. It remains to check that the proposed solution satisfies all the conditions in Theorem 9. It remains only to check the maximum condition. So we have found an optimal solution. The maximum principle Theorem 9. Consequently p is strictly decreasing. The only possible solution is spelled out in the answer in the book. The rest is easy.
Recall that x T is free. It remains to determine p t. For the summing up see the book. The algebra required is heavy. Thus p t is strictly decreasing. We have found the only possible solution. See Note 9. Then from Problem 9. Note that no discounting is assumed, so waiting costs nothing.
The constants A and B are determined from the boundary conditions, and we get the same solution as in Problem 9. Moreover, w is a constant. With the assumptions in Example 8. This implies that H t, A, u, p is concave in A, u. In the answer in the book we go a little further. We differentiate the expression for p t in i w. Conditions i — iv in the answer to Problem 9.
The rest is routine, see the answer in the book. Again the Arrow condition is satisfied, and it is valid also in this case. We use Theorem 9.
Note that the scrap value function is not to be included in the Hamiltonian. A different scrap value function would usually have given another solution. It remains to check that the conditions in iii are satisfied. We have therefore found the solution. Then B and C in Note 9. The Arrow concavity condition holds in this problem and this yields optimality also in the infinite horizon case.
The conditions in iii are obviously satisfied, so we have found the optimal solution. The answer in the book is wrong. For such x t , condition iii is evidently satisfied.
Then condition iii is satisfied and we have found the optimal solution. Taking ln of each side and differentiating w. The first equation is part of the problem. For sufficient conditions, see Note 9. It remains to prove that condition d in Theorem 9. There are also two control variables u1 and u2. The Hamiltonian and the differential equations for p1 and p2 are the same as in a.
The transversality conditions are changed. In fact, according to Theorem In some cases there may be more than one maximum point, but even then at least one corner will be a maximum point.
In the chains of equivalences below it is understood that either the top inequality holds all the way or the bottom inequality holds all the way. The Hamiltonian is concave in x1 , x2 , u , so according to Theorem The other cases are much simpler. The graph of p is shown in Figure We shall use Theorem This is not the optimal solution. In the terminology of Example Finally, using As in Example Note that using the interpretation in Example Thus there exists an optimal control.
We have found the optimal solution. It remains to find p t. The answer is summed up in the answer in the book. Note the misprint in line 3 of the answer to Problem The conditions in Theorem By using this relationship you will see that the values of A and B are the same as in the answer in the book.
The Lagrangian According to Theorem The solution is summed up in the answer in the book. The Hamiltonian is concave in fact linear in x, u and h1 and h2 are quasiconcave in fact linear in x, u.
We claim moreover that p t must be 1. It is a useful exercise to check that all the conditions in i — vii are satisfied. The final conclusion is given in the answer in the book. Throughout the text, large numbers of new and insightful examples and an extensive use of graphs explain and motivate the material. Each chapter develops from an elementary level and builds to more advanced topics, providing logical progression for the student, and enabling instructors to prescribe material to the required level of the course.
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