Similar concepts apply to a set with more than two elements, in which c distinguish between ordered and unordered triples, quadruples, quintuples, and so forth, Ordered pairs, triples, etc. Then 19, and ,19 would obviously mean different things. Moreover, the latter ordered pair would hardly fit any student anywhere. Example 2. When we speak of the set of the five finalists in a contest, the order in which they are listed is of no consequence and we have an unordered quintuple.
But after they are judged, respectively, as the winner, first runner-up, etc. Consider the rectangular cartesian coordinate plane in Fig. This xy plane is an infinite set of points. Clearly, the point labeled 4, 2 is different from the point 2.
With this visual understanding, we are ready to consider the process of generation of ordered pairs. The result will, of course, be the set of four ordered pairs 1,3 , 1,4 , 2,3 , and 2,4. It is important to remember that, while x and y are sets of numbers, the cartesian product turns out to be a set of ordered pairs. Besides, each ordered pair corresponds to a unique point in the cartesian coordinate plane of Fig.
In view of this double uniqueness, a one-to-one correspondence is said to exist between the set of ordered pairs in the cartesian product 2. The rationale for the notation x X y is now easy to perceive; we may associate it with the crossing of the x axis and the y axis in Fig. A simpler way of expressing the set x x y in 2.
Extending this idea, we may also define the cartesian product of three sets x. Furthermore, if the sets x, y. Given an x value, one or more y values will be specified by that relation. For convenience, we shall now write the elements of x x y generally as x, y —rather than as a,b , as was done in 2.
Example 4 The set x. The superscript indicators of the powers of x are called exponents The highest power involved, i. Also, even though we have put the symbol y on the left, itis also acceptable to write f x in its place When plotted in the coordinate plane, a linear function will appear as a straight line, as illustrated in Fig. The other coefficient, a, measures the slope the steepness of incline of our line.
This means that a unit increase in x will result in an increment in y in the amount of a, What Fig. These functions will be used quite frequently in the economic models discussed below. Rational Functions A function such as wernt in which y is expressed as a ratio of two polynomials in the variable x, is known as a rational function again, meaning ratio-nal. According to this definition, any polynomial function must itself be a rational function, because it can always be expressed as a ratio to 1, which is a constant function, A special rational function that has interesting applications in economics is the function a OO ed which plots as a rectangular hyperbola, as in Fig, 2.
Since the product of the wo variables is always a fixed constant in this case, this function may be used to represent that special demand curve—with price and quantity Q on the two axes—for which the total expenditure PQ is constant at all levels of price. Such a demand curve is the one with a unitary elasticity at each point on the curve. Another application is to the average fixed cost AFC curve. With AFC on one axis and output Q on the other. These two types of function will be explained in detail in Chap.
Other types of nonalgebraic function are the rrigonometric or circular functions, which we shall discuss in Chap. We should add here that nonalgebraic func- tions are also known by the more esoteric name of transcendental functions A Digression on Exponents In discussing polynomial functions, we introduced the term exponents as indica tors of the power to which a variable or number is to be raised.
The expression 6? Misbahul Islam. Memo Rojo. Benjamin Franklin. Kevin Hongdi Wang. Judy Domingo. Sabin Sadaf. Renata Lemos. Mauricio Meza Ponze. Syed Asim Bukhari. Anonymous b0cKPzDMqf. Jhon Ortega Garcia. Schaum's Introduction to Mathematical Economics -- Larry Looser. Syed Muzammil Ali. More From Peter G. Peter G. Popular in Internet. Jorge Jarero. R R Baita. Vishal kl. Geovanny Sanjuan. Abul Hasnat. Haftamu Hailu. Megilliandi Boy Alexander.
Zamirtalent Tr. An elephant and a bull sharing a common head. Airavatesvara Temple, India. Giano Bellona. Mahesh Kamdey. Raul Thomas. John Flip. Adeel Syed. Anna M. 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. The zero can be made the last instead of the first element in the principal diagonal, with g1 , g2 and g3 in that order appearing in the last column and in the last row. Both f x and g x are monotonic, and thus quasiconcave.
The function is neither quasiconcave nor quasiconvex. In the x1 x2 plane, this plots for each value of k as a log curve shifted upward vertically by the amount of k.
Thus the function is quasiconvex. Then, by Hence the function is quasiconvex by Hence utility is maximized. Utility is maximized. These check with the preceding problem. These answers check with the preceding problem. The statement is not valid. From Yes, they are true: 4. The elasticity of substitution is merely the elasticity of this line, which can be read by the method of Fig. On the basis of The isoquants would be downward- sloping straight lines.
Consequently, the summation expression can be zero if and only if every component term is zero. This is why the one-equation condition is equivalent to the m separate conditions taken together as a set.
The expanded version of No qualifying arc can be found for this vector. The constraint border is a circle with a radius of 1, and with its center at 0, 0. The optimal solution is at 1, 0. By Thus the test vectors can only point towards due north, northwest, or due west. There does exist a qualifying arc for each such vector. The optimal solution is at the point of origin, a cusp. Moreover, Thus the test vectors must be horizontal, and pointing eastward except for the null vector which does not point anywhere.
Qualifying arcs clearly do exist for each such vector. The curve refers to the graph of the integrand f x. To browse Academia. Sorry, preview is currently unavailable. You can download the paper by clicking the button above. Skip to main content. By using our site, you agree to our collection of information through the use of cookies.
Download books for free. Find books. Download For this fourth edition of a text for students of economics, Chiang University of Connecticut and Wainwright British Columbia Institute of Technology add new chapters on the envelope theorem, advanced topics in optimization, and optimal control theory, and delete a chapter on mathematical Alpha C. Alpha C. All of the textbooks are strongly recommended as useful mathematics books for the duration of the PhD program.
In one determined motion, he grabbed the seal. A hundred thousand paper flowers popped open in the air. My son, Tung Chih, is likely to be the next one. Chiang, Kevin.
0コメント