Table Of ContentSpringer Texts in Business and Economics
Krzysztof Malaga
Karolina Sobczak
Microeconomics
Static and Dynamic Analysis
Springer Texts in Business
and Economics
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Krzysztof Malaga · Karolina Sobczak
Microeconomics
Static and Dynamic Analysis
KrzysztofMalaga KarolinaSobczak
Poznan´UniversityofEconomics Poznan´UniversityofEconomics
andBusiness andBusiness
Poznan´,Poland Poznan´,Poland
ISSN2192-4333 ISSN2192-4341 (electronic)
SpringerTextsinBusiness andEconomics
ISBN978-3-031-10553-1 ISBN978-3-031-10554-8 (eBook)
https://doi.org/10.1007/978-3-031-10554-8
Translation from the Polish language edition: “Mikroekonomia—Uje˛cie statyczne i dynamiczne”
byKrzysztofMalagaandKarolinaSobczak,©ThePublisher2020.PublishedbyC.H.Beck.AllRights
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Contents
1 Introduction ......................................................... 1
2 RationalityofChoicesMadebyIndividualConsumer ............... 9
2.1 Preliminary Terms ............................................... 10
2.2 Utility Function ................................................. 18
2.3 Substitute, Independent and Complementary Goods .............. 30
2.4 Marshallian Demand Function ................................... 42
2.4.1 Static Approach .......................................... 42
2.4.2 Dynamic Approach ....................................... 67
2.5 Hicksian Demand Function ...................................... 75
2.5.1 Static Approach .......................................... 75
2.5.2 Dynamic Approach ....................................... 90
2.6 Substitution and Income Effects of Changes in Prices
of Goods ........................................................ 97
2.6.1 Static Approach .......................................... 97
2.6.2 Dynamic Approach ....................................... 101
2.7 Questions ....................................................... 105
2.8 Exercises ........................................................ 107
3 RationalityofChoicesMadebyGroupofConsumers ............... 113
3.1 Simple Model of Exchange ...................................... 114
3.2 Static Arrow-Hurwicz Model .................................... 122
3.3 Dynamic Arrow-Hurwicz Model ................................. 136
3.4 Questions ....................................................... 155
3.5 Exercises ........................................................ 156
4 RationalityofChoicesMadebyIndividualProducers ............... 159
4.1 Production Space and Production Function ....................... 161
4.2 Substitutability and Complementarity of Production Factors ...... 165
4.3 Financial and Technological Aspects of a Firm’s Activity ........ 169
4.4 Firm Acting in Perfect Competition—Long-Term Strategy ....... 175
4.4.1 Static Approach .......................................... 175
4.4.2 Dynamic Approach ....................................... 193
v
vi Contents
4.5 Firm Acting in Perfect Competition—Short-Term Strategy ....... 202
4.5.1 Static Approach .......................................... 202
4.5.2 Dynamic Approach ....................................... 224
4.6 Monopoly—Long-Term Strategy ................................. 232
4.6.1 Static Approach .......................................... 232
4.6.2 Dynamic Approach ....................................... 247
4.7 Monopoly—Short-Term Strategy ................................ 255
4.7.1 Static Approach .......................................... 255
4.7.2 Dynamic Approach ....................................... 272
4.8 Questions ....................................................... 280
4.9 Exercises ........................................................ 281
5 Rationality of Choices Made by a Group of Producers
byExogenouslyDeterminedFunctionofDemandforaProduct ..... 289
5.1 Firm Acting in Perfect Competition—Determining Optimal
Output Level .................................................... 290
5.2 Monopoly—Determining Optimal Price and Optimal Output
Level ........................................................... 305
5.2.1 Static Approach .......................................... 305
5.2.2 Dynamic Approach ....................................... 318
5.3 Monopolistic Discriminatory Pricing on Two Different
Markets ......................................................... 326
5.3.1 Static Approach .......................................... 326
5.3.2 Dynamic Approach ....................................... 341
5.4 Quantitative and Price Competition of Producers
in a Duopoly .................................................... 348
5.4.1 Cournot Duopoly Model and Its Equilibrium State ........ 349
5.4.2 Stackelberg Duopoly Model and Its Equilibrium State .... 367
5.4.3 Comparative Analysis of Cournot and Stackelberg
Duopoly Models ......................................... 383
5.4.4 Bertrand Duopoly Model and Its Equilibrium State ....... 389
5.5 Questions ....................................................... 407
5.6 Exercises ........................................................ 408
6 Rationality of Choices Made by a Group of Producers
andConsumers ...................................................... 417
6.1 Static Market Model with Exogenous Functions of Supply
and Demand .................................................... 418
6.2 Dynamic Market Model of Two Goods with Exogenous
Functions of Supply and Demand ................................ 423
6.3 Static Arrow-Debreu-McKenzie Model .......................... 428
6.4 Dynamic Arrow-Debreu-McKenzie Model ....................... 438
6.5 Questions ....................................................... 448
6.6 Exercises ........................................................ 449
Contents vii
7 Conclusion ........................................................... 453
MathematicalAppendix ................................................. 459
References ............................................................... 483
Glossary ................................................................. 485
Index .................................................................... 505
Mathematical Symbols
∀ Universal quantifier: “for all”
∃ Existential quantifier: “there exists”
∃ Existential quantifier: “there exists
1
exactly one”
∃>1 Existential quantifier: “there exists more
than one”
¬ Negation; ¬P “not P”
∧ Conjuction: “and”
∨ Disjunction: “or”
⇒ Implication; p ⇒q “if p then q”
⇔ Equivalence: “if and only if”
a,x ∈R Numbers, scalars
a=(a1,a2,...,an)∈Rn+ Vector of parameters
x=(x1,x2,...,xn)∈Rn+ Vector of variables
A,X Sets
A,X Matrices (with dimensions m by) with
m rows and n columns
∼ Indifference relation
x∼y Vector x is indifferent with respect to
vector y
> Strong preference relation
x > y Vector x is strongly preferred over
vector y
> (weak) Preference relation
x>y Vector x is (weakly) preferred over
vector y
R Set of real numbers
R+ Set of nonnegative real numbers
int R+ Set of positive real numbers
Rn =R×R×...×R n-Dimensional space of real numbers,
Cartesian product of set R
Rn+ ={x∈Rn|x≥0}⊂Rn Nonnegative orthant (subspace)
of set Rn
int Rn+ ⊂Rn+ Interior of set Rn
ix
x MathematicalSymbols
f : X →Y Function,
X—Domain of a function (set of
arguments of a function)
Y—Codomain of a function (set of
values of a function)
y = f(x) Scalar function of one variable
x—Independent variable
(argument/input of a function)
y—Independent variable (value of
function)
y = f(x ,x ) Scalar function of two variables
1 2
x ,x —Independent variables
1 2
(arguments/inputs of a function)
y—Independent variable (value of
function)
dy, y", f"(x) Derivative of first order of function
dx
y = f(x)
d2y, y"", f""(x) Derivative of second order of function
dx2
y = f(x)
∂y, ∂f(x1,x2) Partial derivatives of first order for
∂xi ∂xi function y = f(x ,x ) with respect to
1 2
variable x ,i =1,2
i
∂2f(x1,x2), ∂2(x1,x2), i =1,2 Partial derivatives of second order for
∂xi∂xj ⎡∂xi2 ⎤ function y = f(x1,x2),
∂2f(x1,x2) ∂2f(x1,x2)
H(x1,x2)=⎣∂2f∂(xx112,x2) ∂2∂fx(1x∂1x,2x2)⎦ Hessian, symmetric matrix of partial
∂x2∂x1 ∂x22
derivatives of 2nd order for function
| y = f(x1,x2)
|
df(x)| Value of a derivative of one variable
dx x=x¯
| function at point x¯
|
∂f(x1,x2)| Value of a derivative of one variable
∂xi x=x¯
function at point x¯ =(x¯ ,x¯ )
1 2
∑2
<p,x>= pixi Scalar product of two vectors p,x∈R2+
i=1
det(A) or |A| Determinant of matrix A
( )
( ) ∑2 ( ) 21
d x1,x2 = x1−x2 2 Euclidean metric (distance)
E i i
( ) i={1| |}
d x1,x2 = max |x1−x2| Non-Euclidean metric (distance)
i=1,2 i i
( )
1
∑2 2
||x ||= (x )2 Euclidean norm
E i
i=1
