Fundamental Methods of Mathematical Economics

by Wainwright, Kevin; Chiang, Alpha

• ISBN13: 9780070109100
• ISBN10: 0070109109
• Edition: 4th
• Format: Hardcover
• Copyright: 2005-02-02
• Publisher: McGraw Hill

Summary

It has been 20 years since the last edition of this classic text. Kevin Wainwright, a long time user of the text (British Columbia University and Simon Fraser University), has executed the perfect revision-he has updated examples, applications and theory without changing the elegant, precise presentation style of Alpha Chiang. Readers will find the wait was worthwhile.

Table of Contents

PART ONE INTRODUCTION	1	(29)
The Nature of Mathematical Economics	2	(3)
Mathematical versus Nonmathematical Economics	2	(2)
Mathematical Economics versus Econometrics	4	(1)
Economic Models	5	(24)
Ingredients of a Mathematical Model	5	(2)
Variables, Constants, and Parameters	5	(1)
Equations and Identities	6	(1)
The Real-Number System	7	(1)
The Concept of Sets	8	(7)
Set Notation	9	(1)
Relationships between Sets	9	(2)
Operations on Sets	11	(1)
Laws of Set Operations	12	(2)
Exercise 2.3	14	(1)
Relations and Functions	15	(5)
Ordered Pairs	15	(1)
Relations and Functions	16	(3)
Exercise 2.4	19	(1)
Types of Function	20	(5)
Constant Functions	20	(1)
Polynomial Functions	20	(1)
Rational Functions	21	(2)
Nonalgebraic Functions	23	(1)
A Digression on Exponents	23	(1)
Exercise 2.5	24	(1)
Functions of Two or More Independent Variables	25	(2)
Levels of Generality	27	(2)
PART TWO STATIC (OR EQUILIBRIUM) ANALYSIS	29	(94)
Equilibrium Analysis in Economics	30	(18)
The Meaning of Equilibrium	30	(1)
Partial Market Equilibrium---A Linear Model	31	(4)
Constructing the Model	31	(2)
Solution by Elimination of Variables	33	(1)
Exercise 3.2	34	(1)
Partial Market Equilibrium---A Nonlinear Model	35	(5)
Quadratic Equation versus Quadratic Function	35	(1)
The Quadratic Formula	36	(1)
Another Graphical Solution	37	(1)
Higher-Degree Polynomial Equations	38	(2)
Exercise 3.3	40	(1)
General Market Equilibrium	40	(6)
Two-Commodity Market Model	41	(1)
Numerical Example	42	(1)
n-Commodity Case	43	(1)
Solution of a General-Equation System	44	(1)
Exercise 3.4	45	(1)
Equilibrium in National-Income Analysis	46	(2)
Exercise 3.5	47	(1)
Linear Models and Matrix Algebra	48	(34)
Matrices and Vectors	49	(2)
Matrices as Arrays	49	(1)
Vectors as Special Matrices	50	(1)
Exercise 4.1	51	(1)
Matrix Operations	51	(8)
Addition and Subtraction of Matrices	51	(1)
Scalar Multiplication	52	(1)
Multiplication of Matrices	53	(3)
The Question of Division	56	(1)
The Σ Notation	56	(2)
Exercise 4.2	58	(1)
Notes on Vector Operations	59	(8)
Multiplication of Vectors	59	(1)
Geometric Interpretation of Vector Operations	60	(2)
Linear Dependence	62	(1)
Vector Space	63	(2)
Exercise 4.3	65	(2)
Commutative, Associative, and Distributive Laws	67	(3)
Matrix Addition	67	(1)
Matrix Multiplication	68	(1)
Exercise 4.4	69	(1)
Identity Matrices and Null Matrices	70	(3)
Identity Matrices	70	(1)
Null Matrices	71	(1)
Idiosyncrasies of Matrix Algebra	72	(1)
Exercise 4.5	72	(1)
Transposes and Inverses	73	(5)
Properties of Transposes	74	(1)
Inverses and Their Properties	75	(2)
Inverse Matrix and Solution of Linear-Equation System	77	(1)
Exercise 4.6	78	(1)
Finite Markov Chains	78	(4)
Special Case: Absorbing Markov Chains	81	(1)
Exercise 4.7	81	(1)
Linear Models and Matrix Algebra (Continued)	82	(41)
Conditions for Nonsingularity of a Matrix	82	(6)
Necessary versus Sufficient Conditions	82	(2)
Conditions for Nonsingularity	84	(1)
Rank of a Matrix	85	(2)
Exercise 5.1	87	(1)
Test of Nonsingularity by Use of Determinant	88	(6)
Determinants and Nonsingularity	88	(1)
Evaluating a Third-Order Determinant	89	(2)
Evaluating an nth-Order Determinant by Laplace Expansion	91	(2)
Exercise 5.2	93	(1)
Basic Properties of Determinants	94	(5)
Determinantal Criterion for Nonsingularity	96	(1)
Rank of a Matrix Redefined	97	(1)
Exercise 5.3	98	(1)
Finding the Inverse Matrix	99	(4)
Expansion of a Determinant by Alien Cofactors	99	(1)
Matrix Inversion	100	(2)
Exercise 5.4	102	(1)
Cramer's Rule	103	(4)
Derivation of the Rule	103	(2)
Note on Homogeneous-Equation Systems	105	(1)
Solution Outcomes for a Linear-Equation System	106	(1)
Exercise 5.5	107	(1)
Application to Market and National-Income Models	107	(5)
Market Model	107	(1)
National-Income Model	108	(1)
IS-LM Model: Closed Economy	109	(2)
Matrix Algebra versus Elimination of Variables	111	(1)
Exercise 5.6	111	(1)
Leontief Input-Output Models	112	(8)
Structure of an Input-Output Model	112	(1)
The Open Model	113	(2)
A Numerical Example	115	(1)
The Existence of Nonnegative Solutions	116	(2)
Economic Meaning of the Hawkins-Simon Condition	118	(1)
The Closed Model	119	(1)
Exercise 5.7	120	(1)
Limitations of Static Analysis	120	(3)
PART THREE COMPARATIVE-STATIC ANALYSIS	123	(96)
Comparative Statics and the Concept of Derivative	124	(24)
The Nature of Comparative Statics	124	(1)
Rate of Change and the Derivative	125	(3)
The Difference Quotient	125	(1)
The Derivative	126	(1)
Exercise 6.2	127	(1)
The Derivative and the Slope of a Curve	128	(1)
The Concept of Limit	129	(7)
Left-Side Limit and Right-Side Limit	129	(1)
Graphical Illustrations	130	(1)
Evaluation of a Limit	131	(2)
Formal View of the Limit Concept	133	(2)
Exercise 6.4	135	(1)
Digression on Inequalities and Absolute Values	136	(3)
Rules of Inequalities	136	(1)
Absolute Values and Inequalities	137	(1)
Solution of an Inequality	138	(1)
Exercise 6.5	139	(1)
Limit Theorems	139	(2)
Theorems Involving a Single Function	139	(1)
Theorems Involving Two Functions	140	(1)
Limit of a Polynomial Function	141	(1)
Exercise 6.6	141	(1)
Continuity and Differentiability of a Function	141	(7)
Continuity of a Function	141	(1)
Polynomial and Rational Functions	142	(1)
Differentiability of a Function	143	(3)
Exercise 6.7	146	(2)
Rules of Differentiation and Their Use in Comparative Statics	148	(30)
Rules of Differentiation for a Function of One Variable	148	(4)
Constant-Function Rule	148	(1)
Power-Function Rule	149	(2)
Power-Function Rule Generalized	151	(1)
Exercise 7.1	152	(1)
Rules of Differentiation Involving Two or More Functions of the Same Variable	152	(9)
Sum-Difference Rule	152	(3)
Product Rule	155	(1)
Finding Marginal-Revenue Function from Average-Revenue Function	156	(2)
Quotient Rule	158	(1)
Relationship Between Marginal-Cost and Average-Cost Functions	159	(1)
Exercise 7.2	160	(1)
Rules of Differentiation Involving Functions of Different Variables	161	(4)
Chain Rule	161	(2)
Inverse-Function Rule	163	(2)
Exercise 7.3	165	(1)
Partial Differentiation	165	(5)
Partial Derivatives	165	(1)
Techniques of Partial Differentiation	166	(1)
Geometric Interpretation of Partial Derivatives	167	(1)
Gradient Vector	168	(1)
Exercise 7.4	169	(1)
Applications to Comparative-Static Analysis	170	(5)
Market Model	170	(2)
National-Income Model	172	(1)
Input-Output Model	173	(2)
Exercise 7.5	175	(1)
Note on Jacobian Determinants	175	(3)
Exercise 7.6	177	(1)
Comparative-Static Analysis of General-Function Models	178	(41)
Differentials	179	(5)
Differentials and Derivatives	179	(2)
Differentials and Point Elasticity	181	(3)
Exercise 8.1	184	(1)
Total Differentials	184	(3)
Exercise 8.2	186	(1)
Rules of Differentials	187	(2)
Exercise 8.3	189	(1)
Total Derivatives	189	(5)
Finding the Total Derivative	189	(2)
A Variation on the Theme	191	(1)
Another Variation on the Theme	192	(1)
Some General Remarks	193	(1)
Exercise 8.4	193	(1)
Derivatives of Implicit Functions	194	(11)
Implicit Functions	194	(2)
Derivatives of Implicit Functions	196	(3)
Extension to the Simultaneous-Equation Case	199	(5)
Exercise 8.5	204	(1)
Comparative Statics of General-Function Models	205	(13)
Market Model	205	(2)
Simultaneous-Equation Approach	207	(2)
Use of Total Derivatives	209	(1)
National-Income Model (IS-LM)	210	(3)
Extending the Model: An Open Economy	213	(3)
Summary of the Procedure	216	(1)
Exercise 8.6	217	(1)
Limitations of Comparative Statics	218	(1)
PART FOUR OPTIMIZATION PROBLEMS	219	(224)
Optimization: A Special Variety of Equilibrium Analysis	220	(35)
Optimum Values and Extreme Values	221	(1)
Relative Maximum and Minimum: First-Derivative Test	222	(5)
Relative versus Absolute Extremum	222	(1)
First-Derivative Test	223	(3)
Exercise 9.2	226	(1)
Second and Higher Derivatives	227	(6)
Derivative of a Derivative	227	(2)
Interpretation of the Second Derivative	229	(2)
An Application	231	(1)
Attitudes toward Risk	231	(2)
Exercise 9.3	233	(1)
Second-Derivative Test	233	(9)
Necessary versus Sufficient Conditions	234	(1)
Conditions for Profit Maximization	235	(3)
Coefficients of a Cubic Total-Cost Function	238	(2)
Upward-Sloping Marginal-Revenue Curve	240	(1)
Exercise 9.4	241	(1)
Maclaurin and Taylor Series	242	(8)
Maclaurin Series of a Polynomial Function	242	(2)
Taylor Series of a Polynomial Function	244	(1)
Expansion of an Arbitrary Function	245	(3)
Lagrange Form of the Remainder	248	(2)
Exercise 9.5	250	(1)
Nth-Derivative Test for Relative Extremum of a Function of One Variable	250	(5)
Taylor Expansion and Relative Extremum	250	(1)
Some Specific Cases	251	(2)
Nth-Derivative Test	253	(1)
Exercise 9.6	254	(1)
Exponential and Logarithmic Functions	255	(36)
The Nature of Exponential Functions	256	(4)
Simple Exponential Function	256	(1)
Graphical Form	256	(1)
Generalized Exponential Function	257	(2)
A Preferred Base	259	(1)
Exercise 10.1	260	(1)
Natural Exponential Functions and the Problem of Growth	260	(7)
The Number e	260	(2)
An Economic Interpretation of e	262	(1)
Interest Compounding and the Function Aert	262	(1)
Instantaneous Rate of Growth	263	(2)
Continuous versus Discrete Growth	265	(1)
Discounting and Negative Growth	266	(1)
Exercise 10.2	267	(1)
Logarithms	267	(5)
The Meaning of Logarithm	267	(1)
Common Log and Natural Log	268	(1)
Rules of Logarithms	269	(2)
An Application	271	(1)
Exercise 10.3	272	(1)
Logarithmic Functions	272	(5)
Log Functions and Exponential Functions	272	(1)
The Graphical Form	273	(1)
Base Conversion	274	(2)
Exercise 10.4	276	(1)
Derivatives of Exponential and Logarithmic Functions	277	(5)
Log-Function Rule	277	(1)
Exponential-Function Rule	278	(1)
The Rules Generalized	278	(2)
The Case of Base b	280	(1)
Higher Derivatives	280	(1)
An Application	281	(1)
Exercise 10.5	282	(1)
Optimal Timing	282	(4)
A Problem of Wine Storage	282	(1)
Maximization Conditions	283	(2)
A Problem of Timber Cutting	285	(1)
Exercise 10.6	286	(1)
Further Applications of Exponential and Logarithmic Derivatives	286	(5)
Finding the Rate of Growth	286	(1)
Rate of Growth of a Combination of Functions	287	(1)
Finding the Point Elasticity	288	(2)
Exercise 10.7	290	(1)
The Case of More than One Choice Variable	291	(56)
The Differential Version of Optimization Conditions	291	(2)
First-Order Condition	291	(1)
Second-Order Condition	292	(1)
Differential Conditions versus Derivative Conditions	293	(1)
Extreme Values of a Function of Two Variables	293	(8)
First-Order Condition	294	(1)
Second-Order Partial Derivatives	295	(2)
Second-Order Total Differential	297	(1)
Second-Order Condition	298	(2)
Exercise 11.2	300	(1)
Quadratic Forms---An Excursion	301	(12)
Second-Order Total Differential as a Quadratic Form	301	(1)
Positive and Negative Definiteness	302	(1)
Determinantal Test for Sign Definiteness	302	(3)
Three-Variable Quadratic Forms	305	(2)
n-Variable Quadratic Forms	307	(1)
Characteristic-Root Test for Sign Definiteness	307	(5)
Exercise 11.3	312	(1)
Objective Functions with More than Two Variables	313	(5)
First-Order Condition for Extremum	313	(1)
Second-Order Condition	313	(3)
n-Variable Case	316	(1)
Exercise 11.4	317	(1)
Second-Order Conditions in Relation to Concavity and Convexity	318	(13)
Checking Concavity and Convexity	320	(4)
Differentiable Functions	324	(3)
Convex Functions versus Convex Sets	327	(3)
Exercise 11.5	330	(1)
Economic Applications	331	(11)
Problem of a Multiproduct Firm	331	(2)
Price Discrimination	333	(3)
Input Decisions of a Firm	336	(5)
Exercise 11.6	341	(1)
Comparative-Static Aspects of Optimization	342	(5)
Reduced-Form Solutions	342	(1)
General-Function Models	343	(2)
Exercise 11.7	345	(2)
Optimization with Equality Constraints	347	(55)
Effects of a Constraint	347	(2)
Finding the Stationary Values	349	(7)
Lagrange-Multiplier Method	350	(2)
Total-Differential Approach	352	(1)
An Interpretation of the Lagrange Multiplier	353	(1)
n-Variable and Multiconstraint Cases	354	(1)
Exercise 12.2	355	(1)
Second-Order Conditions	356	(8)
Second-Order Total Differential	356	(1)
Second-Order Conditions	357	(1)
The Bordered Hessian	358	(3)
n-Variable Case	361	(1)
Multiconstraint Case	362	(1)
Exercise 12.3	363	(1)
Quasiconcavity and Quasiconvexity	364	(10)
Geometric Characterization	364	(1)
Algebraic Definition	365	(3)
Differentiable Functions	368	(3)
A Further Look at the Bordered Hessian	371	(1)
Absolute versus Relative Extrema	372	(2)
Exercise 12.4	374	(1)
Utility Maximization and Consumer Demand	374	(9)
First-Order Condition	375	(1)
Second-Order Condition	376	(2)
Comparative-Static Analysis	378	(3)
Proportionate Changes in Prices and Income	381	(1)
Exercise 12.5	382	(1)
Homogeneous Functions	383	(7)
Linear Homogeneity	383	(3)
Cobb-Douglas Production Function	386	(2)
Extensions of the Results	388	(1)
Exercise 12.6	389	(1)
Least-Cost Combination of Inputs	390	(12)
First-Order Condition	390	(2)
Second-Order Condition	392	(1)
The Expansion Path	392	(2)
Homothetic Functions	394	(2)
Elasticity of Substitution	396	(1)
CES Production Function	397	(2)
Cobb-Douglas Function as a Special Case of the CES Function	399	(2)
Exercise 12.7	401	(1)
Further Topics in Optimization	402	(41)
Nonlinear Programming and Kuhn-Tucker Conditions	402	(10)
Step 1: Effect of Nonnegativity Restrictions	403	(1)
Step 2: Effect of Inequality Constraints	404	(4)
Interpretation of the Kuhn-Tucker Conditions	408	(1)
The n-Variable, m-Constraint Case	409	(2)
Exercise 13.1	411	(1)
The Constraint Qualification	412	(6)
Irregularities at Boundary Points	412	(3)
The Constraint Qualification	415	(1)
Linear Constraints	416	(2)
Exercise 13.2	418	(1)
Economic Applications	418	(6)
War-Time Rationing	418	(2)
Peak-Load Pricing	420	(3)
Exercise 13.3	423	(1)
Sufficiency Theorems in Nonlinear Programming	424	(4)
The Kuhn-Tucker Sufficiency Theorem: Concave Programming	424	(1)
The Arrow-Enthoven Sufficiency Theorem: Quasiconcave Programming	425	(1)
A Constraint-Qualification Test	426	(1)
Exercise 13.4	427	(1)
Maximum-Value Functions and the Envelope Theorem	428	(7)
The Envelope Theorem for Unconstrained Optimization	428	(1)
The Profit Function	429	(1)
Reciprocity Conditions	430	(2)
The Envelope Theorem for Constrained Optimization	432	(2)
Interpretation of the Lagrange Multiplier	434	(1)
Duality and the Envelope Theorem	435	(7)
The Primal Problem	435	(1)
The Dual Problem	436	(1)
Duality	436	(1)
Roy's Identity	437	(1)
Shephard's Lemma	438	(3)
Exercise 13.6	441	(1)
Some Concluding Remarks	442	(1)
PART FIVE DYNAMIC ANALYSIS	443	(212)
Economic Dynamics and Integral Calculus	444	(31)
Dynamics and Integration	444	(2)
Indefinite Integrals	446	(8)
The Nature of Integrals	446	(1)
Basic Rules of Integration	447	(1)
Rules of Operation	448	(3)
Rules Involving Substitution	451	(2)
Exercise 14.2	453	(1)
Definite Integrals	454	(7)
Meaning of Definite Integrals	454	(1)
A Definite Integral as an Area under a Curve	455	(3)
Some Properties of Definite Integrals	458	(2)
Another Look at the Indefinite Integral	460	(1)
Exercise 14.3	460	(1)
Improper Integrals	461	(3)
Infinite Limits of Integration	461	(2)
Infinite Integrand	463	(1)
Exercise 14.4	464	(1)
Some Economic Applications of Integrals	464	(7)
From a Marginal Function to a Total Function	464	(1)
Investment and Capital Formation	465	(3)
Present Value of a Cash Flow	468	(2)
Present Value of a Perpetual Flow	470	(1)
Exercise 14.5	470	(1)
Domar Growth Model	471	(4)
The Framework	471	(1)
Finding the Solution	472	(1)
The Razor's Edge	473	(1)
Exercise 14.6	474	(1)
Continuous Time: First-Order Differential Equations	475	(28)
First-Order Linear Differential Equations with Constant Coefficient and Constant Term	475	(4)
The Homogeneous Case	476	(1)
The Nonhomogeneous Case	476	(2)
Verification of the Solution	478	(1)
Exercise 15.1	479	(1)
Dynamics of Market Price	479	(4)
The Framework	480	(1)
The Time Path	480	(1)
The Dynamic Stability of Equilibrium	481	(1)
An Alternative Use of the Model	482	(1)
Exercise 15.2	483	(1)
Variable Coefficient and Variable Term	483	(3)
The Homogeneous Case	484	(1)
The Nonhomogeneous Case	485	(1)
Exercise 15.3	486	(1)
Exact Differential Equations	486	(6)
Exact Differential Equations	486	(1)
Method of Solution	487	(2)
Integrating Factor	489	(1)
Solution of First-Order Linear Differential Equations	490	(1)
Exercise 15.4	491	(1)
Nonlinear Differential Equations of the First Order and First Degree	492	(3)
Exact Differential Equations	492	(1)
Separable Variables	492	(1)
Equations Reducible to the Linear Form	493	(2)
Exercise 15.5	495	(1)
The Qualitative-Graphic Approach	495	(3)
The Phase Diagram	495	(1)
Types of Time Path	496	(2)
Exercise 15.6	498	(1)
Solow Growth Model	498	(5)
The Framework	498	(2)
A Qualitative-Graphic Analysis	500	(1)
A Quantitative Illustration	501	(1)
Exercise 15.7	502	(1)
Higher-Order Differential Equations	503	(41)
Second-Order Linear Differential Equations with Constant Coefficients and Constant Term	504	(7)
The Particular Integral	504	(1)
The Complementary Function	505	(5)
The Dynamic Stability of Equilibrium	510	(1)
Exercise 16.1	511	(1)
Complex Numbers and Circular Functions	511	(11)
Imaginary and Complex Numbers	511	(1)
Complex Roots	512	(1)
Circular Functions	513	(2)
Properties of the Sine and Cosine Functions	515	(2)
Euler Relations	517	(2)
Alternative Representations of Complex Numbers	519	(2)
Exercise 16.2	521	(1)
Analysis of the Complex-Root Case	522	(5)
The Complementary Function	522	(2)
An Example of Solution	524	(1)
The Time Path	525	(2)
The Dynamic Stability of Equilibrium	527	(1)
Exercise 16.3	527	(1)
A Market Model with Price Expectations	527	(5)
Price Trend and Price Expectations	527	(1)
A Simplified Model	528	(1)
The Time Path of Price	529	(3)
Exercise 16.4	532	(1)
The Interaction of Inflation and Unemployment	532	(6)
The Phillips Relation	532	(1)
The Expectations-Augmented Phillips Relation	533	(1)
The Feedback from Inflation to Unemployment	534	(1)
The Time Path of π	534	(3)
Exercise 16.5	537	(1)
Differential Equations with a Variable Term	538	(2)
Method of Undetermined Coefficients	538	(1)
A Modification	539	(1)
Exercise 16.6	540	(1)
Higher-Order Linear Differential Equations	540	(4)
Finding the Solution	540	(2)
Convergence and the Routh Theorem	542	(1)
Exercise 16.7	543	(1)
Discrete Time: First-Order Difference Equations	544	(24)
Discrete Time, Differences, and Difference Equations	544	(2)
Solving a First-Order Difference Equation	546	(5)
Iterative Method	546	(2)
General Method	548	(3)
Exercise 17.2	551	(1)
The Dynamic Stability of Equilibrium	551	(4)
The Significance of b	551	(2)
The Role of A	553	(1)
Convergence to Equilibrium	554	(1)
Exercise 17.3	554	(1)
The Cobweb Model	555	(4)
The Model	555	(1)
The Cobwebs	556	(2)
Exercise 17.4	558	(1)
A Market Model with Inventory	559	(3)
The Model	559	(1)
The Time Path	560	(1)
Graphical Summary of the Results	561	(1)
Exercise 17.5	562	(1)
Nonlinear Difference Equations---The Qualitative-Graphic Approach	562	(6)
Phase Diagram	562	(2)
Types of Time Path	564	(1)
A Market with a Price Ceiling	565	(2)
Exercise 17.6	567	(1)
Higher-Order Difference Equations	568	(24)
Second-Order Linear Difference Equations with Constant Coefficients and Constant Term	569	(7)
Particular Solution	569	(1)
Complementary Function	570	(3)
The Convergence of the Time Path	573	(2)
Exercise 18.1	575	(1)
Samuelson Multiplier-Acceleration Interaction Model	576	(5)
The Framework	576	(1)
The Solution	577	(1)
Convergence versus Divergence	578	(2)
A Graphical Summary	580	(1)
Exercise 18.2	581	(1)
Inflation and Unemployment in Discrete Time	581	(5)
The Model	581	(1)
The Difference Equation in p	582	(1)
The Time Path of p	583	(1)
The Analysis of U	584	(1)
The Long-Run Phillips Relation	585	(1)
Exercise 18.3	585	(1)
Generalizations to Variable-Term and Higher-Order Equations	586	(6)
Variable Term in the Form of cmt	586	(1)
Variable Term in the Form ctn	587	(1)
Higher-Order Linear Difference Equations	588	(1)
Convergence and the Schur Theorem	589	(2)
Exercise 18.4	591	(1)
Simultaneous Differential Equations and Difference Equations	592	(39)
The Genesis of Dynamic Systems	592	(2)
Interacting Patterns of Change	592	(1)
The Transformation of a High-Order Dynamic Equation	593	(1)
Solving Simultaneous Dynamic Equations	594	(9)
Simultaneous Difference Equations	594	(2)
Matrix Notation	596	(3)
Simultaneous Differential Equations	599	(2)
Further Comments on the Characteristic Equation	601	(1)
Exercise 19.2	602	(1)
Dynamic Input-Output Models	603	(6)
Time Lag in Production	603	(2)
Excess Demand and Output Adjustment	605	(2)
Capital Formation	607	(1)
Exercise 19.3	608	(1)
The Inflation-Unemployment Model Once More	609	(5)
Simultaneous Differential Equations	610	(1)
Solution Paths	610	(2)
Simultaneous Difference Equations	612	(1)
Solution Paths	613	(1)
Exercise 19.4	614	(1)
Two-Variable Phase Diagrams	614	(9)
The Phase Space	615	(1)
The Demarcation Curves	615	(2)
Streamlines	617	(1)
Types of Equilibrium	618	(2)
Inflation and Monetary Rule a la Obst	620	(3)
Exercise 19.5	623	(1)
Linearization of a Nonlinear Differential-Equation System	623	(8)
Taylor Expansion and Linearization	624	(1)
The Reduced Linearization	625	(1)
Local Stability Analysis	625	(4)
Exercise 19.6	629	(2)
Optimal Control Theory	631	(24)
The Nature of Optimal Control	631	(8)
Illustration: A Simple Macroeconomic Model	632	(1)
Pontryagin's Maximum Principle	633	(6)
Alternative Terminal Conditions	639	(5)
Fixed Terminal Point	639	(1)
Horizontal Terminal Line	639	(1)
Truncated Vertical Terminal Line	639	(1)
Truncated Horizontal Terminal Line	640	(3)
Exercise 20.2	643	(1)
Autonomous Problems	644	(1)
Economic Applications	645	(4)
Lifetime Utility Maximization	645	(2)
Exhaustible Resource	647	(2)
Exercise 20.4	649	(1)
Infinite Time Horizon	649	(5)
Neoclassical Optimal Growth Model	649	(2)
The Current-Value Hamiltonian	651	(1)
Constructing a Phase Diagram	652	(1)
Analyzing the Phase Diagram	653	(1)
Limitations of Dynamic Analysis	654	(1)
The Greek Alphabet	655	(1)
Mathematical Symbols	656	(3)
A Short Reading List	659	(3)
Answers to Selected Exercises	662	(15)
Index	677