

1  (29) 

The Nature of Mathematical Economics 


2  (3) 

Mathematical versus Nonmathematical Economics 


2  (2) 

Mathematical Economics versus Econometrics 


4  (1) 


5  (24) 

Ingredients of a Mathematical Model 


5  (2) 

Variables, Constants, and Parameters 


5  (1) 


6  (1) 


7  (1) 


8  (7) 


9  (1) 

Relationships between Sets 


9  (2) 


11  (1) 


12  (2) 


14  (1) 


15  (5) 


15  (1) 


16  (3) 


19  (1) 


20  (5) 


20  (1) 


20  (1) 


21  (2) 


23  (1) 

A Digression on Exponents 


23  (1) 


24  (1) 

Functions of Two or More Independent Variables 


25  (2) 


27  (2) 

PART TWO STATIC (OR EQUILIBRIUM) ANALYSIS 


29  (94) 

Equilibrium Analysis in Economics 


30  (18) 

The Meaning of Equilibrium 


30  (1) 

Partial Market EquilibriumA Linear Model 


31  (4) 


31  (2) 

Solution by Elimination of Variables 


33  (1) 


34  (1) 

Partial Market EquilibriumA Nonlinear Model 


35  (5) 

Quadratic Equation versus Quadratic Function 


35  (1) 


36  (1) 

Another Graphical Solution 


37  (1) 

HigherDegree Polynomial Equations 


38  (2) 


40  (1) 

General Market Equilibrium 


40  (6) 

TwoCommodity Market Model 


41  (1) 


42  (1) 


43  (1) 

Solution of a GeneralEquation System 


44  (1) 


45  (1) 

Equilibrium in NationalIncome Analysis 


46  (2) 


47  (1) 

Linear Models and Matrix Algebra 


48  (34) 


49  (2) 


49  (1) 

Vectors as Special Matrices 


50  (1) 


51  (1) 


51  (8) 

Addition and Subtraction of Matrices 


51  (1) 


52  (1) 

Multiplication of Matrices 


53  (3) 


56  (1) 


56  (2) 


58  (1) 

Notes on Vector Operations 


59  (8) 

Multiplication of Vectors 


59  (1) 

Geometric Interpretation of Vector Operations 


60  (2) 


62  (1) 


63  (2) 


65  (2) 

Commutative, Associative, and Distributive Laws 


67  (3) 


67  (1) 


68  (1) 


69  (1) 

Identity Matrices and Null Matrices 


70  (3) 


70  (1) 


71  (1) 

Idiosyncrasies of Matrix Algebra 


72  (1) 


72  (1) 


73  (5) 


74  (1) 

Inverses and Their Properties 


75  (2) 

Inverse Matrix and Solution of LinearEquation System 


77  (1) 


78  (1) 


78  (4) 

Special Case: Absorbing Markov Chains 


81  (1) 


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) 


85  (2) 


87  (1) 

Test of Nonsingularity by Use of Determinant 


88  (6) 

Determinants and Nonsingularity 


88  (1) 

Evaluating a ThirdOrder Determinant 


89  (2) 

Evaluating an nthOrder Determinant by Laplace Expansion 


91  (2) 


93  (1) 

Basic Properties of Determinants 


94  (5) 

Determinantal Criterion for Nonsingularity 


96  (1) 

Rank of a Matrix Redefined 


97  (1) 


98  (1) 

Finding the Inverse Matrix 


99  (4) 

Expansion of a Determinant by Alien Cofactors 


99  (1) 


100  (2) 


102  (1) 


103  (4) 


103  (2) 

Note on HomogeneousEquation Systems 


105  (1) 

Solution Outcomes for a LinearEquation System 


106  (1) 


107  (1) 

Application to Market and NationalIncome Models 


107  (5) 


107  (1) 


108  (1) 

ISLM Model: Closed Economy 


109  (2) 

Matrix Algebra versus Elimination of Variables 


111  (1) 


111  (1) 

Leontief InputOutput Models 


112  (8) 

Structure of an InputOutput Model 


112  (1) 


113  (2) 


115  (1) 

The Existence of Nonnegative Solutions 


116  (2) 

Economic Meaning of the HawkinsSimon Condition 


118  (1) 


119  (1) 


120  (1) 

Limitations of Static Analysis 


120  (3) 

PART THREE COMPARATIVESTATIC 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) 


125  (1) 


126  (1) 


127  (1) 

The Derivative and the Slope of a Curve 


128  (1) 


129  (7) 

LeftSide Limit and RightSide Limit 


129  (1) 


130  (1) 


131  (2) 

Formal View of the Limit Concept 


133  (2) 


135  (1) 

Digression on Inequalities and Absolute Values 


136  (3) 


136  (1) 

Absolute Values and Inequalities 


137  (1) 

Solution of an Inequality 


138  (1) 


139  (1) 


139  (2) 

Theorems Involving a Single Function 


139  (1) 

Theorems Involving Two Functions 


140  (1) 

Limit of a Polynomial Function 


141  (1) 


141  (1) 

Continuity and Differentiability of a Function 


141  (7) 


141  (1) 

Polynomial and Rational Functions 


142  (1) 

Differentiability of a Function 


143  (3) 


146  (2) 

Rules of Differentiation and Their Use in Comparative Statics 


148  (30) 

Rules of Differentiation for a Function of One Variable 


148  (4) 


148  (1) 


149  (2) 

PowerFunction Rule Generalized 


151  (1) 


152  (1) 

Rules of Differentiation Involving Two or More Functions of the Same Variable 


152  (9) 


152  (3) 


155  (1) 

Finding MarginalRevenue Function from AverageRevenue Function 


156  (2) 


158  (1) 

Relationship Between MarginalCost and AverageCost Functions 


159  (1) 


160  (1) 

Rules of Differentiation Involving Functions of Different Variables 


161  (4) 


161  (2) 


163  (2) 


165  (1) 


165  (5) 


165  (1) 

Techniques of Partial Differentiation 


166  (1) 

Geometric Interpretation of Partial Derivatives 


167  (1) 


168  (1) 


169  (1) 

Applications to ComparativeStatic Analysis 


170  (5) 


170  (2) 


172  (1) 


173  (2) 


175  (1) 

Note on Jacobian Determinants 


175  (3) 


177  (1) 

ComparativeStatic Analysis of GeneralFunction Models 


178  (41) 


179  (5) 

Differentials and Derivatives 


179  (2) 

Differentials and Point Elasticity 


181  (3) 


184  (1) 


184  (3) 


186  (1) 


187  (2) 


189  (1) 


189  (5) 

Finding the Total Derivative 


189  (2) 


191  (1) 

Another Variation on the Theme 


192  (1) 


193  (1) 


193  (1) 

Derivatives of Implicit Functions 


194  (11) 


194  (2) 

Derivatives of Implicit Functions 


196  (3) 

Extension to the SimultaneousEquation Case 


199  (5) 


204  (1) 

Comparative Statics of GeneralFunction Models 


205  (13) 


205  (2) 

SimultaneousEquation Approach 


207  (2) 


209  (1) 

NationalIncome Model (ISLM) 


210  (3) 

Extending the Model: An Open Economy 


213  (3) 


216  (1) 


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: FirstDerivative Test 


222  (5) 

Relative versus Absolute Extremum 


222  (1) 


223  (3) 


226  (1) 

Second and Higher Derivatives 


227  (6) 

Derivative of a Derivative 


227  (2) 

Interpretation of the Second Derivative 


229  (2) 


231  (1) 


231  (2) 


233  (1) 


233  (9) 

Necessary versus Sufficient Conditions 


234  (1) 

Conditions for Profit Maximization 


235  (3) 

Coefficients of a Cubic TotalCost Function 


238  (2) 

UpwardSloping MarginalRevenue Curve 


240  (1) 


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) 


250  (1) 

NthDerivative Test for Relative Extremum of a Function of One Variable 


250  (5) 

Taylor Expansion and Relative Extremum 


250  (1) 


251  (2) 


253  (1) 


254  (1) 

Exponential and Logarithmic Functions 


255  (36) 

The Nature of Exponential Functions 


256  (4) 

Simple Exponential Function 


256  (1) 


256  (1) 

Generalized Exponential Function 


257  (2) 


259  (1) 


260  (1) 

Natural Exponential Functions and the Problem of Growth 


260  (7) 


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) 


267  (1) 


267  (5) 


267  (1) 

Common Log and Natural Log 


268  (1) 


269  (2) 


271  (1) 


272  (1) 


272  (5) 

Log Functions and Exponential Functions 


272  (1) 


273  (1) 


274  (2) 


276  (1) 

Derivatives of Exponential and Logarithmic Functions 


277  (5) 


277  (1) 

ExponentialFunction Rule 


278  (1) 


278  (2) 


280  (1) 


280  (1) 


281  (1) 


282  (1) 


282  (4) 

A Problem of Wine Storage 


282  (1) 


283  (2) 

A Problem of Timber Cutting 


285  (1) 


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) 


290  (1) 

The Case of More than One Choice Variable 


291  (56) 

The Differential Version of Optimization Conditions 


291  (2) 


291  (1) 


292  (1) 

Differential Conditions versus Derivative Conditions 


293  (1) 

Extreme Values of a Function of Two Variables 


293  (8) 


294  (1) 

SecondOrder Partial Derivatives 


295  (2) 

SecondOrder Total Differential 


297  (1) 


298  (2) 


300  (1) 

Quadratic FormsAn Excursion 


301  (12) 

SecondOrder Total Differential as a Quadratic Form 


301  (1) 

Positive and Negative Definiteness 


302  (1) 

Determinantal Test for Sign Definiteness 


302  (3) 

ThreeVariable Quadratic Forms 


305  (2) 

nVariable Quadratic Forms 


307  (1) 

CharacteristicRoot Test for Sign Definiteness 


307  (5) 


312  (1) 

Objective Functions with More than Two Variables 


313  (5) 

FirstOrder Condition for Extremum 


313  (1) 


313  (3) 


316  (1) 


317  (1) 

SecondOrder Conditions in Relation to Concavity and Convexity 


318  (13) 

Checking Concavity and Convexity 


320  (4) 


324  (3) 

Convex Functions versus Convex Sets 


327  (3) 


330  (1) 


331  (11) 

Problem of a Multiproduct Firm 


331  (2) 


333  (3) 

Input Decisions of a Firm 


336  (5) 


341  (1) 

ComparativeStatic Aspects of Optimization 


342  (5) 


342  (1) 


343  (2) 


345  (2) 

Optimization with Equality Constraints 


347  (55) 


347  (2) 

Finding the Stationary Values 


349  (7) 

LagrangeMultiplier Method 


350  (2) 

TotalDifferential Approach 


352  (1) 

An Interpretation of the Lagrange Multiplier 


353  (1) 

nVariable and Multiconstraint Cases 


354  (1) 


355  (1) 


356  (8) 

SecondOrder Total Differential 


356  (1) 


357  (1) 


358  (3) 


361  (1) 


362  (1) 


363  (1) 

Quasiconcavity and Quasiconvexity 


364  (10) 

Geometric Characterization 


364  (1) 


365  (3) 


368  (3) 

A Further Look at the Bordered Hessian 


371  (1) 

Absolute versus Relative Extrema 


372  (2) 


374  (1) 

Utility Maximization and Consumer Demand 


374  (9) 


375  (1) 


376  (2) 

ComparativeStatic Analysis 


378  (3) 

Proportionate Changes in Prices and Income 


381  (1) 


382  (1) 


383  (7) 


383  (3) 

CobbDouglas Production Function 


386  (2) 

Extensions of the Results 


388  (1) 


389  (1) 

LeastCost Combination of Inputs 


390  (12) 


390  (2) 


392  (1) 


392  (2) 


394  (2) 

Elasticity of Substitution 


396  (1) 


397  (2) 

CobbDouglas Function as a Special Case of the CES Function 


399  (2) 


401  (1) 

Further Topics in Optimization 


402  (41) 

Nonlinear Programming and KuhnTucker Conditions 


402  (10) 

Step 1: Effect of Nonnegativity Restrictions 


403  (1) 

Step 2: Effect of Inequality Constraints 


404  (4) 

Interpretation of the KuhnTucker Conditions 


408  (1) 

The nVariable, mConstraint Case 


409  (2) 


411  (1) 

The Constraint Qualification 


412  (6) 

Irregularities at Boundary Points 


412  (3) 

The Constraint Qualification 


415  (1) 


416  (2) 


418  (1) 


418  (6) 


418  (2) 


420  (3) 


423  (1) 

Sufficiency Theorems in Nonlinear Programming 


424  (4) 

The KuhnTucker Sufficiency Theorem: Concave Programming 


424  (1) 

The ArrowEnthoven Sufficiency Theorem: Quasiconcave Programming 


425  (1) 

A ConstraintQualification Test 


426  (1) 


427  (1) 

MaximumValue Functions and the Envelope Theorem 


428  (7) 

The Envelope Theorem for Unconstrained Optimization 


428  (1) 


429  (1) 


430  (2) 

The Envelope Theorem for Constrained Optimization 


432  (2) 

Interpretation of the Lagrange Multiplier 


434  (1) 

Duality and the Envelope Theorem 


435  (7) 


435  (1) 


436  (1) 


436  (1) 


437  (1) 


438  (3) 


441  (1) 


442  (1) 

PART FIVE DYNAMIC ANALYSIS 


443  (212) 

Economic Dynamics and Integral Calculus 


444  (31) 


444  (2) 


446  (8) 


446  (1) 

Basic Rules of Integration 


447  (1) 


448  (3) 

Rules Involving Substitution 


451  (2) 


453  (1) 


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) 


460  (1) 


461  (3) 

Infinite Limits of Integration 


461  (2) 


463  (1) 


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) 


470  (1) 


471  (4) 


471  (1) 


472  (1) 


473  (1) 


474  (1) 

Continuous Time: FirstOrder Differential Equations 


475  (28) 

FirstOrder Linear Differential Equations with Constant Coefficient and Constant Term 


475  (4) 


476  (1) 


476  (2) 

Verification of the Solution 


478  (1) 


479  (1) 


479  (4) 


480  (1) 


480  (1) 

The Dynamic Stability of Equilibrium 


481  (1) 

An Alternative Use of the Model 


482  (1) 


483  (1) 

Variable Coefficient and Variable Term 


483  (3) 


484  (1) 


485  (1) 


486  (1) 

Exact Differential Equations 


486  (6) 

Exact Differential Equations 


486  (1) 


487  (2) 


489  (1) 

Solution of FirstOrder Linear Differential Equations 


490  (1) 


491  (1) 

Nonlinear Differential Equations of the First Order and First Degree 


492  (3) 

Exact Differential Equations 


492  (1) 


492  (1) 

Equations Reducible to the Linear Form 


493  (2) 


495  (1) 

The QualitativeGraphic Approach 


495  (3) 


495  (1) 


496  (2) 


498  (1) 


498  (5) 


498  (2) 

A QualitativeGraphic Analysis 


500  (1) 

A Quantitative Illustration 


501  (1) 


502  (1) 

HigherOrder Differential Equations 


503  (41) 

SecondOrder Linear Differential Equations with Constant Coefficients and Constant Term 


504  (7) 


504  (1) 

The Complementary Function 


505  (5) 

The Dynamic Stability of Equilibrium 


510  (1) 


511  (1) 

Complex Numbers and Circular Functions 


511  (11) 

Imaginary and Complex Numbers 


511  (1) 


512  (1) 


513  (2) 

Properties of the Sine and Cosine Functions 


515  (2) 


517  (2) 

Alternative Representations of Complex Numbers 


519  (2) 


521  (1) 

Analysis of the ComplexRoot Case 


522  (5) 

The Complementary Function 


522  (2) 


524  (1) 


525  (2) 

The Dynamic Stability of Equilibrium 


527  (1) 


527  (1) 

A Market Model with Price Expectations 


527  (5) 

Price Trend and Price Expectations 


527  (1) 


528  (1) 


529  (3) 


532  (1) 

The Interaction of Inflation and Unemployment 


532  (6) 


532  (1) 

The ExpectationsAugmented Phillips Relation 


533  (1) 

The Feedback from Inflation to Unemployment 


534  (1) 


534  (3) 


537  (1) 

Differential Equations with a Variable Term 


538  (2) 

Method of Undetermined Coefficients 


538  (1) 


539  (1) 


540  (1) 

HigherOrder Linear Differential Equations 


540  (4) 


540  (2) 

Convergence and the Routh Theorem 


542  (1) 


543  (1) 

Discrete Time: FirstOrder Difference Equations 


544  (24) 

Discrete Time, Differences, and Difference Equations 


544  (2) 

Solving a FirstOrder Difference Equation 


546  (5) 


546  (2) 


548  (3) 


551  (1) 

The Dynamic Stability of Equilibrium 


551  (4) 


551  (2) 


553  (1) 

Convergence to Equilibrium 


554  (1) 


554  (1) 


555  (4) 


555  (1) 


556  (2) 


558  (1) 

A Market Model with Inventory 


559  (3) 


559  (1) 


560  (1) 

Graphical Summary of the Results 


561  (1) 


562  (1) 

Nonlinear Difference EquationsThe QualitativeGraphic Approach 


562  (6) 


562  (2) 


564  (1) 

A Market with a Price Ceiling 


565  (2) 


567  (1) 

HigherOrder Difference Equations 


568  (24) 

SecondOrder Linear Difference Equations with Constant Coefficients and Constant Term 


569  (7) 


569  (1) 


570  (3) 

The Convergence of the Time Path 


573  (2) 


575  (1) 

Samuelson MultiplierAcceleration Interaction Model 


576  (5) 


576  (1) 


577  (1) 

Convergence versus Divergence 


578  (2) 


580  (1) 


581  (1) 

Inflation and Unemployment in Discrete Time 


581  (5) 


581  (1) 

The Difference Equation in p 


582  (1) 


583  (1) 


584  (1) 

The LongRun Phillips Relation 


585  (1) 


585  (1) 

Generalizations to VariableTerm and HigherOrder Equations 


586  (6) 

Variable Term in the Form of cmt 


586  (1) 

Variable Term in the Form ctn 


587  (1) 

HigherOrder Linear Difference Equations 


588  (1) 

Convergence and the Schur Theorem 


589  (2) 


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 HighOrder Dynamic Equation 


593  (1) 

Solving Simultaneous Dynamic Equations 


594  (9) 

Simultaneous Difference Equations 


594  (2) 


596  (3) 

Simultaneous Differential Equations 


599  (2) 

Further Comments on the Characteristic Equation 


601  (1) 


602  (1) 

Dynamic InputOutput Models 


603  (6) 


603  (2) 

Excess Demand and Output Adjustment 


605  (2) 


607  (1) 


608  (1) 

The InflationUnemployment Model Once More 


609  (5) 

Simultaneous Differential Equations 


610  (1) 


610  (2) 

Simultaneous Difference Equations 


612  (1) 


613  (1) 


614  (1) 

TwoVariable Phase Diagrams 


614  (9) 


615  (1) 


615  (2) 


617  (1) 


618  (2) 

Inflation and Monetary Rule a la Obst 


620  (3) 


623  (1) 

Linearization of a Nonlinear DifferentialEquation System 


623  (8) 

Taylor Expansion and Linearization 


624  (1) 

The Reduced Linearization 


625  (1) 


625  (4) 


629  (2) 


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) 


639  (1) 


639  (1) 

Truncated Vertical Terminal Line 


639  (1) 

Truncated Horizontal Terminal Line 


640  (3) 


643  (1) 


644  (1) 


645  (4) 

Lifetime Utility Maximization 


645  (2) 


647  (2) 


649  (1) 


649  (5) 

Neoclassical Optimal Growth Model 


649  (2) 

The CurrentValue 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  