But the set in b cannot. They are ai3 Ci2 and a2j C4j , respectively. Yes, Matrices G and H in problem 4 are examples. Thus the matrix equation is 0. Element 0: Industry III does not use its own output as its input. Element The open sector demands billion dollars of commodity II. Exercise 6. Yes, the limit is For example, 2. Yes; each function is not only continuous but also smooth.
Refer to the following two graphs Exercise 7. Since the total-cost function shows zero fixed cost, the situation depicted is the long run. Exercise 7. The equations in 7. Hence, by 8. At least one of the partial derivatives in the vector of constants in 8.
Exercise 8. Thus the implicit- function theorem is applicable. A similar complication arises when the supply function is used to get the other comparative-static derivative. These are in the nature of relative extrema, thus a minimum can exceed a maximum.
A Exercise 9. It plots as an upward-sloping straight line emanating from the point of origin. Exercise 9. The critical values are 1 and 5. With its minimum at 3a zero output. The MC curve must be upward-sloping throughout. The value of marginal product must be equated to the wage rate. Thus, by 9. Since the first nonzero derivative value is f 4 2.
The value of t can be either positive, zero, or negative. Exercise The nonzero requirement serves to preclude this contingency. The graphs are of the same general shape as in Fig. Similarly to formula Thus, by Thus Both are the same as before. Both roots being positive, u0 Du is positive definite. Both roots being negative, u0 Eu is negative definite. Since r1 is positive, but r2 is negative, u0 F u is indefinite. Using r1 in The diagonal elements are all negative for problem 2, and all positive for problems 4 and 5.
Thus the characteristic roots are all negative d2 z negative definite for problem 2, and all positive d2 z positive definite for problem 4. Thus the maximum in this problem is a unique absolute maximum. The highest 1 is cd1 ; the lowest is cd2. Letting Pa0 alone vary i. In Thus, those elements are really the second-order partial derivatives of the primitive objective function — exactly what are used to construct the Hessian determinant.
Then it follows that d2 g must be zero, too. Thus d2 z does not have to be zero as a matter of course. Khurmi 1. A: Proofs of Selected Theorems A. B: Integration Tables A. C: Precalculus Review Online A. E: Complex Numbers Online A. Adams — 6th Edition. Essential Calculus — James Stewart — 1st Edition. Multivariable Calculus — James Stewart — 5th Edition. The Classic Edition Vol. Swokowski, Jeffery A. Cole — 1st Edition. Calculus — James Stewart — 6th Edition. Download now Calculus.
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