By Geoffrey Alexander Jehle Philip J. Reny
This rigorous, up to date textual content on sleek microeconomic conception provides the entire middle arithmetic, neoclassical concept, online game conception, and knowledge economics had to entry the fashionable specialist literature. advanced conception is patiently and thoroughly constructed, then essentially defined and illustrated simply because even well-prepared scholars reap the benefits of extra math aid. cautious causes, effective theorem-proof association, and lots of examples and workouts make this a uniquely potent textual content for complex classes. scholars will have fun with the transparent writing and available kind.
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7, y = e(p, 11) > 0. 6, we may choose t > 0 small enough so that y - £ > 0 and v(p, y - B) > 11 . Thus, iDCOme y - t is sufficient, 8l prices p, to achieve utilily greater than u. Hence, we must have e(p, u) � y - e. But this contradicts the fact thai y = e(p, u). I so this says 11 2:: u + t, a conlnldiclion. Hence, e(p, v(p, y)) = y. he other a minimization problem. This theorem, howevec, poinu to an easy way to derive either one from knowledge of the odler, thus requiring us to solve only one optimization problem and giving us the choice of which 011e we care to tions, we would have had to solve two a IIIIWmization problem and 1b see how this would work, l�t's suppose lint that we have solved the utility problem and formed the indirect utilily function.
C�(p. u ) + �>: •�(p, u) = e•. More generally, we define the expeuditure f\IDdion as the minimum-value function, e(p, u) e xEr. t. 11(x) ;::, 11 for all p » 0 and all attainable utility levels 11 . For futute reference, let U = I u(x)ls denote the set of attainable utility levels. Thus. the domain of e(-) is R�+ x U. ( 1 . 14) E R� ) Note that e(p, 11) is well-defined because for p E R�. x E R� . p · x 2: 0. Hence. the set of numben fe I t = p · x for some x with 11(x) ;:: 11) is bounded below by zero.
Then the cost of x 1 at prices p1 must be no more than the cost at prices p1 of any other bundle x that achieves utility u. l the cost at p2 ofo11yother bundle To x that achieves utility 11. Now, if, as we've said, · and for all x that achieve 11 , then these relations must also hold for x•, because x' achieves u as well. Therefore, simply by virtue of what it me1111s to minimize expenditure to achieve 11 at . . -�c CHAPTER 1 given prices, we know that and But now we are home free. Because t � 0 and (I - t) ;:>: 0, we can multiply the tint of these by t, the set4nd by (I - t), and add diem.