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Fundamental Mathematical Economic solutions - Studocu Chiang Fundamental Mathematical Economic solutions - Studocu

S (read: “8 is not an element of set S ”). If we use the symbol R to denote but obviously 8 ∈ the set of all real numbers, then the statement “x is some real number” can be simply expressed by x∈R Chapter 17 Discrete Time: First-Order Difference Equations 544 17.1 Discrete Time, Differences, and Difference Equations 544 17.2 Solving a First-Order Difference Equation 546 Iterative Method 546 General Method 548 Exercise 17.2 551 very good math preparation book for econ major. It could be used to undergraduate text, or graduate review book, since the level is a bit lower. Access-restricted-item true Addeddate 2019-12-11 01:31:32 Boxid IA1736419 Camera USB PTP Class Camera Collection_set printdisabled External-identifier Second-Derivative Test 233 Necessary versus Sufficient Conditions 234 Conditions for Profit Maximization 235 Coefficients of a Cubic Total-Cost Function 238 Upward-Sloping Marginal-Revenue Curve 240 Exercise 9.4 241

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Second-Order Conditions in Relation to Concavity and Convexity 318 Checking Concavity and Convexity 320 Differentiable Functions 324 Convex Functions versus Convex Sets 327 Exercise 11.5 330

Chiang Fundamental Mathematical Economics solution - Academia.edu Chiang Fundamental Mathematical Economics solution - Academia.edu

where Q denotes the quantity of output. Since the two equations have different forms, the production condition assumed in each is obviously different from the other. In (2.1), the fixed cost (the value of C when Q = 0) is 75, whereas in (2.2) it is 110. The variation in cost is also different. In (2.1), for each unit increase in Q, there is a constant increase of 10 in C. But in (2.2), as Q increases unit after unit, C will increase by progressively larger amounts. Clearly, it is primarily through the specification of the form of the behavioral equations that we give mathematical expression to the assumptions adopted for a model. As the third type, a conditional equation states a requirement to be satisfied. For example, in a model involving the notion of equilibrium, we must set up an equilibrium condition, which describes the prerequisite for the attainment of equilibrium. Two of the most familiar equilibrium conditions in economics are The Derivative and the Slope of a Curve 128 6.4 The Concept of Limit 129 Left-Side Limit and Right-Side Limit 129 Graphical Illustrations 130 Evaluation of a Limit 131 Formal View of the Limit Concept 133 Exercise 6.4 135Some Economic Applications of Integrals 464 From a Marginal Function to a Total Function 464 Investment and Capital Formation 465 Present Value of a Cash Flow 468 Present Value of a Perpetual Flow 470 Exercise 14.5 470 Duality and the Envelope Theorem 435 The Primal Problem 435 The Dual Problem 436 Duality 436 Roy’s Identity 437 Shephard’s Lemma 438 Exercise 13.6 441 Differential Equations with a Variable Term 538 Method of Undetermined Coefficients 538 A Modification 539 Exercise 16.6 540 The choice between literary logic and mathematical logic, again, is a matter of little import, but mathematics has the advantage of forcing analysts to make their assumptions explicit at every stage of reasoning. This is because mathematical theorems are usually stated in the “if-then” form, so that in order to tap the “then” (result) part of the theorem for their use, they must first make sure that the “if ” (condition) part does conform to the explicit assumptions adopted. Granting these points, though, one may still ask why it is necessary to go beyond geometric methods. The answer is that while geometric analysis has the important advantage of being visual, it also suffers from a serious dimensional limitation. In the usual graphical discussion of indifference curves, for instance, the standard assumption is that only two commodities are available to the consumer. Such a simplifying assumption is not willingly adopted but is forced upon us because the task of drawing a three-dimensional graph is exceedingly difficult, and the construction of a four- (or higher) dimensional graph is actually a physical impossibility. To deal with the more general case of 3, 4, or n goods, we must instead resort to the more flexible tool of equations. This reason alone should provide sufficient motivation for the study of mathematical methods beyond geometry. In short, we see that the mathematical approach has claim to the following advantages: (1) The “language” used is more concise and precise; (2) there exists a wealth of mathematical theorems at our service; (3) in forcing us to state explicitly all our assumptions as a prerequisite to the use of the mathematical theorems, it keeps us from the pitfall of an unintentional adoption of unwanted implicit assumptions; and (4) it allows us to treat the general n-variable case. Against these advantages, one sometimes hears the criticism that a mathematically derived theory is inevitably unrealistic. However, this criticism is not valid. In fact, the epithet “unrealistic” cannot even be used in criticizing economic theory in general, whether or not the approach is mathematical. Theory is by its very nature an abstraction from the real world. It is a device for singling out only the most essential factors and relationships so that we can study the crux of the problem at hand, free from the many complications that do exist in the actual world. Thus the statement “theory lacks realism” is merely a truism that cannot be accepted as a valid criticism of theory. By the same token, it is quite meaningless to pick out any one approach to theory as “unrealistic.” For example, the theory of firm under pure competition is unrealistic, as is the theory of firm under imperfect competition, but whether these theories are derived mathematically or not is irrelevant and immaterial. To take advantage of the wealth of mathematical tools, one must of course first acquire those tools. Unfortunately, the tools that are of interest to economists are widely scattered among many mathematics courses—too many to fit comfortably into the plan of study of a typical economics student. The service the present volume performs is to gather in one place the mathematical methods most relevant to the economics literature, organize them into a logical order of progression, fully explain each method, and then immediately illustrate how the method is applied in economic analysis. By tying together the methods and their applications, the relevance of mathematics to economics is made more transparent than is possible in the regular mathematics courses where the illustrated applications are predominantly tied to physics and engineering. Familiarity with the contents of this book (and, if possible, also its sequel volume: Alpha C. Chiang, Elements of Dynamic Optimization, McGraw-Hill, 1992, now published by Waveland Press, Inc.) should therefore enable you to comprehend most of the professional articles you will come across in

Fundamental Methods of Mathematical Economics Solution Manual for Fundamental Methods of Mathematical Economics

Second and Higher Derivatives 227 Derivative of a Derivative 227 Interpretation of the Second Derivative 229 An Application 231 Attitudes toward Risk 231 Exercise 9.3 233 Samuelson Multiplier-Acceleration Interaction Model 576 The Framework 576 The Solution 577 Convergence versus Divergence 578 A Graphical Summary 580 Exercise 18.2 581 Economic Dynamics and Integral Calculus 444 15 Continuous Time: First-Order Differential Equations 475 16 Higher-Order Differential Equations 503 17 Discrete Time: First-Order Difference Equations 544 18 Higher-Order Difference Equations 568 19 Simultaneous Differential Equations and Difference Equations 592 20 Optimal Control Theory 631 Dynamic Input-Output Models 603 Time Lag in Production 603 Excess Demand and Output Adjustment 605 Capital Formation 607 Exercise 19.3 608

Objective Functions with More than Two Variables 313 First-Order Condition for Extremum 313 Second-Order Condition 313 n-Variable Case 316 Exercise 11.4 317 Chapter 12 Optimization with Equality Constraints 347 12.1 Effects of a Constraint 347 12.2 Finding the Stationary Values 349 Lagrange-Multiplier Method 350 Total-Differential Approach 352 An Interpretation of the Lagrange Multiplier 353 n-Variable and Multiconstraint Cases 354 Exercise 12.2 355