Solutions for Foundations of Mathematical Economics fundamental result that all bases have the same number of elements Therefore, x′ is a lower cost method of producing one unit of output, contradicting the. Instructor s Manual and Solutions Manual Chiang A.C., Wainwright K. Solutions Manual for Fundamental Methods of Mathematical Economics It has been Chiang/Wainwright: Fundamental Methods of Mathematical Economics Instructor’s Manual CHAPTER 2 Exercise 1 Chiang/Wainwright: Multiplying both sides of the latter by 1/4, we get the solution −3/4 < x. (b) The.
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Chiang Fundamental Mathematical Economics solution | Carlos Javier Solis Herrera –
Chiang Fundamental Mathematical Economics solution. Only d represents a function. The range is the set of all nonpositive numbers. This would establish the uniqueness as required by the definition of a function. Being negative, this term serves to pull down the y values at the two extreme ends of the curve. If negative values can occur there will appear in quadrant III a curve which is the mirror image of the one in quadrant I. The two curves would be parallel, with no equilibrium intersection point in Fig.
By Theorem III, we find: The resulting equation is the same as the one in b above. Y, C, and T. No, because the equation system is nonlinear 4. When u, v, and w all lie on a single straight line. Let the vector v have the elements a1. The point of origin has the elements 0. No, x0 Ax would then contain cross-product terms a12 x1 x2 and a21 x1 x2. Unweighted sum of squares is used in the well-known method of least squares for fitting an equation to a set of data.
Weighted sum of squares can be used, e. The given diagonal matrix, when multiplied by itself, gives another diagonal matrix with the diagonal elements aa. Hence each aii must be either 1, or 0. Since each aii can thus have two possible values, and since there are altogether n of these aiiwe are able to construct a total of 2n idempotent matrices of the diagonal type.
Two examples would be In and 0n. It is suggested that this particular problem could also be solved using a spreadsheet or other mathematical software. The student will be able to observe features of a Markov process more quickly without doing the repetitive calculations. We get the same results as in the preceding problem.
In A1 keep row 1 as is, but add row 1 to row 2, to get A2. In A2divide row 2 by 5. In Fundzmentaldivide row 1 by 6. In B2multiply row 2 by 2, then add the new row 2 to row 3. There mathematocal linear dependence in B: In C1 divide row 1 to 7. The question of nonsingularity is not relevant here because C is not square. But the interchange of rows 1 and 2 gives us simpler numbers to work with. Since the last two rows of D2are identical, linear dependence is obvious. Again, the question nonsingularity is not relevant here.
The link is provided by the third elementary row operation. This process involves the third elementary row operation. Factoring out the k in each successive column or row —for a total of n columns or rows —will yield the indicated result. In mathematjcalthe rank is 3. In ab and cthe rank is less than 3. The set in fundmental can because when the three vectors are combined into a matrix, its determinant does not vanish.
But the set in b cannot.
They are ai3 Ci2 and a2j C4jrespectively. Yes, Matrices G and H in problem 4 are examples. After the indicated multiplication by the appropriate cofactors, the new equations will add up to the following equation: Thus the matrix equation is 0. Industry III does not use its own output as its input.
The open sector demands billion dollars of commodity II. Yes, the limit is 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. The equations in 7.
Thus an implicit function is defined, with: At least one of the partial derivatives in the vector of constants in 8. Thus the implicit- function theorem is applicable. The equilibrium identity is: A similar complication arises when the supply function is used to get the other comparative-static derivative.
Since the Fundamenntal is nonzero: These are in the nature of relative extrema, thus a minimum can exceed a maximum. To use it as a consumption function, we should c stipulate that: It plots as an upward-sloping straight line emanating from the point of origin.
Chiang A.C., Wainwright K. Solutions Manual for Fundamental Methods of Mathematical Economics
The critical values are 1 and 5. With its minimum at 3a zero output. The MC curve must be econoics throughout. The value of marginal product must be equated to the wage rate. Since the first nonzero derivative value is f 4 2. The value of t can be either positive, zero, or negative. The nonzero requirement serves to preclude this contingency. The graphs are of the same general shape as in Fig.
Similarly to formula 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 One way of doing it is as follows: The diagonal elements are all negative for problem 2, and all positive for problems 4 and 5. Thus the characteristic roots are all fundamnetal 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.
Thus, those elements are really the second-order partial derivatives of the primitive objective function — exactly what are used to construct the Hessian determinant. The necessary condition is: The first-order condition becomes: Then it follows that d2 g must be zero, too. Thus d2 z does not have to be zero as a matter of course. The zero can be made the last instead of the first element in the principal diagonal, with g1g2 and g3 in that order appearing in the last column and in the last row.
Examples of acceptable curves are: Both f x and g x are monotonic, mathemmatical thus quasiconcave. The function is neither quasiconcave nor quasiconvex. In the x1 x2 plane, this plots for each value of k funeamental a log curve shifted upward vertically by the amount of k.
Thus the function is quasiconvex. Hence the function is quasiconvex by Hence utility is maximized.
These check with the preceding problem. These answers check with the preceding problem. The statement is not valid.