By Geoffrey Alexander Jehle Philip J. Reny
This rigorous, updated textual content on sleek microeconomic thought provides all the center arithmetic, neoclassical concept, online game concept, and knowledge economics had to entry the trendy expert literature. complicated idea is patiently and thoroughly constructed, then basically defined and illustrated simply because even well-prepared scholars reap the benefits of extra math support. cautious reasons, effective theorem-proof association, and lots of examples and routines make this a uniquely powerful textual content for complicated classes. scholars will savor the transparent writing and available variety.
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Additional info for Advanced Microeconomic Theory
Our concern is with an individual consumer operating within a market economy. 8. 2. lMAClOH ECOIOii1CA 20 Flpre U. Blldgec set, 8 = Jx i • E R� . p · ll � )'), in the case of two commoditi�s. by markets. et for each commodity, and in these markets, a price p; prevails for each commodity i. We suppose that prices are strictly positive, so p1 > 0, i = I, . . , n . Moreover, w e assume the individual consumer is an insignifiCtJ/11 force on every market. By this we mean, specifically, that the size of each martel relative to the potential puc chases of the individual consumer is so large that no ma� bow much or how little the consumer might purchase, there will be no percepcible effect on any marlcet price.
Under our additional hypotheses, we can now use the Envelope theorem to show thai t(p, u) is strictly increas By the Envelope theorem, the panial derivative ofthe minimum-value function e(p, u) ing in u. wilb respect to u is equal to the partial derivative of the Lagrangian wilh respect to u , evaluated Ill (x • . • ). •) = }.. au > . O Because this holds for all u > 11(0), and because tO is continuous, we may conclude that for all p » 0, t(p, u) is strictly iRCrefSing in 11 on U (which includes u(O)).
Consequently, at prices p, if the consumer's income were in fact y. then he could attain at least the level of utility u. Because 11(p, y) is the wrrest utility level attainable at prices p and with income y, this implies that 11(p, y) � u. Consequently, the definitions of v and t also imply that u(p, e(p, u)) � 11 V(p, u) E R:+ x U. ( 1 . scrates that under certain familiar conditions on prefuences, both of these inequalities, in fact, musr be equaliries. 8 Ler u(p, y) and e(p, u) be the indirtct utility function and uptnditurt jiUICtion for 1omt consiUflt'r wltost utilityfunctimt il colllinuous and llrictly itrcrtaslng.
Advanced Microeconomic Theory by Geoffrey Alexander Jehle Philip J. Reny