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Unformatted text preview: Jehle_Chapter May 18, 2011 15:5 # 631 H INTS AND A NSWERS C HAPTER 1 1.2 Use the definitions. 1.4 To get you started, take the indifference relation. Consider any three points x i 2 X ; i D 1; 2; 3; where x 1 x 2 and x 2 x 3 . We want to show that x 1 x 2 and x 2 x 3 ) x 1 x 3 . By definition of ; x 1 x 2 ) x 1 % x 2 and x 2 % x 1 . Similarly, x 2 x 3 ) x 2 % x 3 and x 3 % x 2 . By transitivity of % ; x 1 % x 2 and x 2 % x 3 ) x 1 % x 3 . Keep going. 1.16 For (a), suppose there is some other feasible bundle x , where x x . Use the fact that B is convex, together with strict convexity of preferences, to derive a contradiction. For (b), suppose not. Use strict monotonicity to derive a contradiction. 1.22 Use a method similar to that employed in (1.11) to eliminate the Lagrangian multiplier and reduce . n C 1/ conditions to only n conditions. 1.23 For part (2), see Axiom 5 : Note that the sets % . x / are precisely the superior sets for the function u . x / . Recall Theorem A1.14. 1.27 Sketch out the indifference map. 1.28 For part (a), suppose by way of contradiction that the derivative is negative. 1.29 Set down all firstorder conditions. Look at the one for choice of x . Use the constraint, and find a geometric series. Does it converge? 1.32 Feel free to assume that any necessary derivatives exist. 1.33 Roy’s identity. 1.41 Theorem A2.6. 1.46 Euler’s theorem and any demand function, x i . p ; y / . 1.47 For part (a), start with the definition of e . p ; 1/ . Multiply the constraint by u and invoke homogeneity. Let z u x and rewrite the objective function as a choice over z . Jehle_Chapter May 18, 2011 15:5 # 632 632 HINTS AND ANSWERS 1.52 Take each inequality separately. Write the one as @ x i . p i ; y /=@ y x i . p ; y / N y : Integrate both sides of the inequality from N y to y and look for logs. Take it from there. 1.54 For part (b), v . p ; y / D A y n Y i D 1 p ˛ i i ; where A D A Q n i D 1 ˛ ˛ i i . 1.60 Use Slutsky. 1.63 No hints on this. 1.66 For (b), u must be v . p ; y / , right? Rewrite the denominator. 1.67 For (a), you need the expenditure function and you need to figure out u . For (b), I D . u 1=8/= .2 u 1/ . For (c), if you could show that the expenditure function must be multiplicatively separable in prices and utility, the rest would be easy. C HAPTER 2 2.3 It should be a CobbDouglas form. 2.9 Use a diagram. 2.10 To get you started, x 2 is revealed preferred to x 1 . 2.12 For (a), use GARP to show that, unless . x j / is zero, there is a minimising sequence of distinct numbers k 1 ; :::; k m defining . x j / such that no k 1 ; :::; k m is equal to j : Hence, k 1 ; :::; k m ; j is a feasible sequence for the minimisation problem defining . x k / . For (b), use (a). For (c), recall that each p k 2 R n CC : For (e), the minimum of quasiconcave functions is quasiconcave....
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 Spring '11
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