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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 first-order 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 /[email protected] 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 Cobb-Douglas 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
- Monotonic function, Convex function, X2 X1, x1 x2