Contents
List of illustrations page xiii Introduction to the second edition xv Preface to the second editionxxi Preface to the first edition xxiii
Table of notation xxv
Table of assumptions xxix
A General equilibrium theory: Getting acquainted 1
1 Concept and history of general equilibrium theory 3
1.1 Partial and general equilibrium: Development of the field 3
1.2 The role of mathematics 7
1.3 History of general equilibrium theory 8
1.4 Bibliographic note 10
2 An elementary general equilibrium model: The Robinson Crusoe
economy 12
2.1 Centralized allocation14
2.2 Decentralized allocation 16
2.3 Pareto efficiency of the competitive equilibrium allocation: First fundamental Theorem of Welfare Economics 23
2.4 Bibliographic note 24
Exercises 24
3 The Edgeworth box 31
3.1 Geometry of the Edgeworth box 32
3.2 Calculating an efficient allocation 35
3.3 A competitive market solution in the Edgeworth box 37
3.4 Bibliographic note 40
Exercises 40
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viii Contents
4 Integrating production and multiple consumption decisions:
A2×2×2 model 44
4.1 A 2×2×2 model46
4.2 Technical efficiency46 4.3 Pareto efficiency 48
4.4 First Fundamental Theorem of Welfare Economics:
Competitive equilibrium is Pareto efficient 50
Exercises 52
5 Existence of general equilibrium in an economy with an excess
demand function 58
5.1 Bibliographic note 64
Exercises 64
B Mathematics 67
6 Logic and set theory69
6.1 Quasi-orderings 71
6.2 Functions 73
6.3 Bibliographic note 73
Exercises 73
7 RN: Real N-dimensional Euclidean space 75
7.1 Continuous functions82
7.2 Bibliographic note 85
Exercises 85
8 Convex sets, separation theorems, and nonconvex sets in RN 91
8.1 Separation theorems 92
8.2 The Shapley-Folkman Theorem 95
8.3 Bibliographic note 97
Exercises 98
9 The Brouwer Fixed-Point Theorem 99
9.1 Bibliographic note 106
Exercises 106
C An economy with bounded production technology and supply
and demand functions 109
10 Markets, prices, commodities, and mathematical economic
theory 112
10.1 Commodities and prices 112
10.2 The formal structure of pure economic theory 112
10.3 Markets, commodities, and prices 113
10.4 Bibliographic note 114
Exercise 114
Contents ix
11 Production with bounded-firm technology 115
11.1 Firms and production technology 115
11.2 The form of production technology 116
11.3 Strictly convex production technology 117
11.4 Aggregate supply 120
11.5 Attainable production plans 120
11.6 Bibliographic note 121
Exercises 121
12 Households 124
12.1 The structure of household consumption sets and preferences 124
12.2 Consumption sets 124
12.3 Representation ofi: Existence of a continuous utility function 129
12.4 Choice and boundedness of budget sets, ˜
Bi(p) 131
12.5 Demand behavior under strict convexity 134
12.6 Bibliographic note 137
Exercises137
13 A market economy 142
13.1 Firms, profits, and household income 142
13.2 Excess demand and Walras’s Law 143
13.3 Bibliographic note 145
Exercises 145
14 General equilibrium of the market economy with an excess
demand function 147
14.1 Existence of equilibrium 147
14.2 Bibliographic note 152
Exercises 152
D An economy with unbounded production technology and
supply and demand functions 161
15 Theory of production: The unbounded technology case 164
15.1 Unbounded production technology 164
15.2 Boundedness of the attainable set 165
15.3 An artificially bounded supply function 169
15.4 Bibliographic note 172
Exercises 172
16 Households: The unbounded technology case 174
16.1 Households 174
16.2 Choice in an unbounded budget set 174
16.3 Demand behavior under strict convexity 177
16.4 Bibliographic note 179
Exercise 179
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17 A market economy: The unbounded technology case 180
17.1 Firms and households 180
17.2 Profits 180
17.3 Household income 181
17.4 Excess demand and Walras’s Law 181
17.5 Bibliographic note 184
Exercises 184
18 General equilibrium of the market economy: The unbounded
technology case 185
18.1 General equilibrium 185
18.2 An artificially restricted economy 186
18.3 General equilibrium of the unrestricted economy 187
18.4 The Uzawa Equivalence Theorem 190
18.5 Bibliographic note 193
Exercises 193
E Welfare economics and the scope of markets 203
19 Pareto efficiency and competitive equilibrium 205
19.1 Pareto efficiency 205
19.2 First Fundamental Theorem of Welfare Economics 206
19.3 Second Fundamental Theorem of Welfare Economics 209
19.4 Corner solutions214
19.5 Bibliographic note 214
Exercises 214
20 Time and uncertainty: Futures markets 225
20.1 Introduction 225
20.2 Time: Futures markets 227
20.3 Uncertainty: Arrow-Debreu contingent commodity markets 233
20.4 Uncertainty: Arrow securities markets 238
20.5 Conclusion: The missing markets 241
20.6 Bibliographic note 242
Exercises 243
F Bargaining and equilibrium: The core 249
21 The core of a market economy 251
21.1 Bargaining and competition 251
21.2 The core of a pure exchange economy 252
21.3 The competitive equilibrium allocation is in the core 254
21.4 Bibliographic note 255
Exercise 255
Contents xi
22 Convergence of the core of a large economy 256
22.1 Replication: A large economy 256
22.2 Equal treatment 257
22.3 Core convergence in a large economy 259
22.4 A large economy without replication 263
22.5 Interpreting the core convergence result 267
22.6 Bibliographic note 268
Exercises 269
G An economy with supply and demand correspondences 275
23 Mathematics: Analysis of point-to-set mappings 279
23.1 Correspondences 279
23.2 Upper hemicontinuity (also known as upper semicontinuity) 279
23.3 Lower hemicontinuity (also known as lower semicontinuity) 282
23.4 Continuous correspondence 284
23.5 Cartesian product of correspondences 285
23.6 Optimization subject to constraint: Composition of
correspondences; the Maximum Theorem 285
23.7 Kakutani Fixed-Point Theorem 287
23.8 Bibliographic note 291
Exercises 291
24 General equilibrium of the market economy with an excess demand
correspondence 293
24.1 General equilibrium with set-valued supply and demand 293
24.2 Production with a (weakly) convex production technology 294
24.3 Households 298
24.4 The market economy 304
24.5 The artificially restricted economy 307
24.6 Existence of competitive equilibrium 308
24.7 Bibliographic note 310
Exercises 310
25 U-shaped cost curves and concentrated preferences 312
25.1 U-shaped cost curves and concentrated preferences 312
25.2 The nonconvex economy 313
25.3 Artificial convex counterpart to the nonconvex economy 314
25.4 Approximate equilibrium 317
25.5 Bibliographic note 319
Exercises 320
xii Contents
H Standing on the shoulders of giants 323
26 Next steps 325
26.1 Large economies325
26.2 Anything goes! 327
26.3 Regular economies and the determinacy of equilibrium 328
26.4 General equilibrium with incomplete markets 329
26.5 Computing general equilibrium 330
26.6 Bibliographic note331
27 Summary and conclusion 332
27.1 Overview and summary 332
27.2 Bibliographic note 333
Exercises 334
Bibliography 335
Index 341
List of illustrations
2.1 The Robinson Crusoe economy: Efficient allocation. page 14
2.2 The Robinson Crusoe economy: Equilibrium and disequilibrium. 17
3.1 The Edgeworth box. 32
3.2 The Edgeworth box: Bargaining and allocation. 33
3.3 The Edgeworth box: Efficient allocation and the contract curve. 37
3.4 The Edgeworth box: Disequilibrium. 39
3.5 The Edgeworth box: General equilibrium. 39
4.1 A two-good economy: General equilibrium in production and
distribution.45
7.1 A vector in R2.76
7.2 Vector addition. 77
8.1 Convex and nonconvex sets. 92
8.2 Bounding and separating hyperplanes for convex sets. 94
9.1 The Brouwer Fixed-Point Theorem in R. 100
9.2 An admissibly labeled simplicial subdivision of a simplex. 101 9.3 Sperner’s Lemma for N = 1.102
11.1 Yj: Technology set of firm j. 116
11.2 Convex and nonconvex technology sets. 118
12.1 Lexicographic preferences.128
14.1 Mapping from P into P. 149
15.1 Bounding firm j’s production technology. 169
16.1 Household i’s budget sets and demand functions. 178
18.1 The Uzawa Equivalence Theorem. 192
19.1 Supporting an efficient allocation (Theorem 19.2). 210
20.1 Uncertain states of the world: An event tree. 234
22.1 Core convergence (Theorem 22.2). 261
22.2 Nonconvex preferences (Exercise 22.6). 272
G.1 Linear production technology and its supply correspondence. 276 G.2 Preferences for perfect substitutes and the demand correspondence. 277
G.3 Equilibrium in a market with supply and demand correspondences. 278
23.1 A typical correspondence, ϕ(x) ={y|x −1 ≤ y ≤ x +1}. 280
23.2 Example 23.1 – An upper hemicontinuous correspondence. 281
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xiv List of illustrations
23.3 Example 23.2 – A correspondence that is not upper hemicontinuous at 0. 282
23.4 Example 23.3 – A lower hemicontinuous correspondence. 283
23.5 Example 23.5 – A continuous correspondence. 284
23.6 The maximum problem. 286
23.7 An upper hemicontinuous mapping from an interval (1-simplex) into
itself without a fixed point. 288
23.8 An upper hemicontinuous convex-valued mapping from an interval
(1-simplex) into itself with a fixed point. 288
23.9 Lemma 23.2 – Approximating an upper hemicontinuous convex-valued
correspondence by a continuous function. 289
23.10 Example 23.7 – Applying the Kakutani Fixed-Point Theorem. 290
24.1 Example 24.1 – An upper hemicontinuous, convex-valued supply
correspondence. 295
24.2 Example 23.2 – An upper hemicontinuous supply correspondence that is
not convex valued. 296
24.3 Theorem 23.2 – Continuity of the budget set showing the construction
of yν. 301
26.1 An economy with an infinite number of equilibria. 328
