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2 Hierarchic Preferences 9
2.1 Basic Set-up ............................... 9
2.1.1 Individual Demand .................. ..... 12
2.1.2 Aggregate Demand ............. .......... 14
2.2 Hierarchy and CRRA ...................... ... 20
2.3 Non-Homothetic Utility in the Previous Theoretical Literature . 23
2.4 Appendix ..... . ........... ............. . . 26
2.4.1 Convergence of the Utility Integral . . . . . . . . . . . ... 26
2.4.2 Proof of Proposition 1 ..................... 28
3 Structural Change and Balanced Growth 31
3.1 Introduction . . . . . . . . .... . . . . . . . . . . . . 31
3.2 Static Equilibrium ...... ............. ..... 36
3.2.1 Preferences and Demand . . . . . . . . . . . . . . ..... . 36
3.2.2 Optimal Price Setting ..................... 37
3.2.3 Demand and Price Structure . . . . . . . . . . . . . .... 37
3.3 Dynamics of the Economy . . . . . . . . . . . . . . ..... . . . . 40
3.3.1 R&D and the Resource Constraint ........ . ...... .. 40
3.3.2 Intertemporal Consumption Choice . . . . . . . . . . ... 42
3.4 Equilibrium Growth Path ..... ................... 43
3.4.1 Definition of Equilibrium Growth Path . . . . . . . .. . . . 43
3.4.2 A Unique Equilibrium ..................... 45
3.5 Structural Change and the Kaldor Facts . . . . . . . . . . . . ... 49
3.5.1 The Kaldor Facts ........................ 49
3.5.2 Structural Change ....................... 51
3.5.3 The Impact of Hierarchic Preferences . . . . . . . . . ... 55
3.6 Calibration ................... . .......... 57
3.7 Other Types of Equilibria ....................... 61
3.8 Is the Optimal Patent Duration Infinite? . . . . . . . . . . . . ... 64
3.9 Discussion and Extensions . . . . . . . . . . . . . . ..... . . . . 66
3.10 Appendix ................... . .......... .. 68
3.10.1 Proof of Lemma 1 ....................... 68
3.10.2 Proof of Lemma 2 ................... ..... 68
3.10.3 Proof of Lemma 3 ....................... 70
3.10.4 Proof of Proposition 4 .... ................. 71
3.10.5 Proof of Proposition 9 (Patent Duration) . . . . . . . ... 71
4 Inequality and Research Incentives 75
4.1 Introduction . . . . . . . . . . . . . . ..... . . . . . . . . .. .... 75
4.2 Static Equilibrium ................... ........ 78
4.2.1 Hierarchic Preferences and Consumption Choices . . . . . . 78
4.2.2 The Determination of Prices and the Structure of Consump-
tion ....... ............ ............ 80
4.2.3 Solving the Static Consumers' Problem . . . . . . . . ... 83
4.2.4 Static Expenditures and Utilities . . . . . . . . . . . . . . . 85
4.3 Dynamics of the Economy . . . . . . . . . . . . . . ..... . . . . 87
4.3.1 Intertemporal Allocation of Expenditures . . . . . . . .. 87
4.3.2 The Supply Side: Technology and Resource Constraint . . . 88
4.3.3 The Innovation Process . . . . . . . . . . . . . . ..... . 91
4.4 Distribution of Income and Wealth . . . . . . . . . . . . . . . ... 93
4.5 General Equilibrium ................... ...... 94
4.5.1 The Three Possible Regimes . . . . . . . . . . . . . . . ... 95
4.5.2 A Graphical Representation of the Equilibrium ...... . . 98
4.5.3 Steeper Hierarchy ....................... 102
4.6 The Impact of Inequality on Growth . . . . . . . . . . . . . . .. 103
4.6.1 No Traditional Sector (v = 0) . . . . . ..... ......... ..103
4.6.2 The General Case v > 0 .................... 104
4.7 Sum m ary . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 106
4.8 A ppendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.8.1 The Case With Preemptive Patenting . . . . . . . . . ... 108
4.8.2 The Case Where All Individuals Buy All Products . . . . . 112
4.8.3 Proof of Lemma 5 ...... ................. 114
4.8.4 Proof of Proposition 11 . . . . . . ..... .......... .. 116
5 Markups and Exclusion 117
5.1 Introduction ................... ............ 117
5.2 Monopolistic Competition With Non-Homothetic Preferences . . . 121
5.2.1 Consumers and Firms ..................... 123
5.2.2 Restrictions on Preferences and Distribution . . . . . .. .124
5.3 Symmetric versus Asymmetric Equilibria . . . . . . . . . . . ... 126
5.3.1 Symmetric Equilibrium . . . . . . . . . . . . . . ..... . 126
5.3.2 A Condition Whether Symmetry Constitutes an Equilibrium 129
5.3.3 The Asymmetric Case ('Exclusion') . . . . ..... . . . . 131
5.4 Unemployment ................... ......... 135
5.4.1 Symmetric Equilibrium . . . . . . . . . . . . . . ..... . 135
5.4.2 Asymmetric Equilibrium . . . . . . . ..... ........ ..136
5.5 Discussion ................... ............ 139
5.6 Appendix ................... ............. 141
6 Conclusions 143