Solution Jehle Reny
Solutions to selected exercises from Jehle and Reny (2001): Advanced Microeconomic Theory
Thomas Herzfeld September 2010 Contents
1 Mathematical Appendix 1.1 Chapter A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Chapter A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Consumer Theory 2.1 Preferences and Utility . . . . . . 2.2 The Consumer’s Problem . . . . . 2.3 Indirect Utility and Expenditure . 2.4 Properties of Consumer Demand 2.5 Equilibrium and Welfare . . . . . 3 Producer Theory 3.1 Production . . . . . 3.2 Cost . . . . . . . . . 3.3 Duality in production 3.4 The competitive firm 2 2 6 12 12 14 16 18 20 23 23 26 28 30
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1 Mathematical Appendix
1 Mathematical Appendix
1.1 Chapter A1
A1.7 Graph each of the following sets. If the set is convex, give a proof. If it is not convex, give a counterexample. Answer (a) (x, y)|y = ex This set is not convex. Any combination of points would be outside the set. For example, (0, 1) and 1 / (1, e) ∈ (x, y)|y = ex , but combination of the two vectors with t = 2 not: ( 1 , e+1 ) ∈ 2 2 x (x, y)|y = e . (b) (x, y)|y ≥ ex This set is convex. Proof: Let (x1 , y1 ), (x2 , y2 ) ∈ S = (x, y)|y ≥ ex . Since y = ex is a continuous function, it is sufficient to show that (tx1 + (1 − t)x2 , ty1 + (1 − t)y2 ) ∈ S for any particular t ∈ (0, 1). Set t = 1 . Our task is to show that 1
Thomas Herzfeld September 2010 Contents
1 Mathematical Appendix 1.1 Chapter A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Chapter A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Consumer Theory 2.1 Preferences and Utility . . . . . . 2.2 The Consumer’s Problem . . . . . 2.3 Indirect Utility and Expenditure . 2.4 Properties of Consumer Demand 2.5 Equilibrium and Welfare . . . . . 3 Producer Theory 3.1 Production . . . . . 3.2 Cost . . . . . . . . . 3.3 Duality in production 3.4 The competitive firm 2 2 6 12 12 14 16 18 20 23 23 26 28 30
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1
1 Mathematical Appendix
1 Mathematical Appendix
1.1 Chapter A1
A1.7 Graph each of the following sets. If the set is convex, give a proof. If it is not convex, give a counterexample. Answer (a) (x, y)|y = ex This set is not convex. Any combination of points would be outside the set. For example, (0, 1) and 1 / (1, e) ∈ (x, y)|y = ex , but combination of the two vectors with t = 2 not: ( 1 , e+1 ) ∈ 2 2 x (x, y)|y = e . (b) (x, y)|y ≥ ex This set is convex. Proof: Let (x1 , y1 ), (x2 , y2 ) ∈ S = (x, y)|y ≥ ex . Since y = ex is a continuous function, it is sufficient to show that (tx1 + (1 − t)x2 , ty1 + (1 − t)y2 ) ∈ S for any particular t ∈ (0, 1). Set t = 1 . Our task is to show that 1
References: Arrow, K. J. & Enthoven, A. C. (1961), ‘Quasi-concave programming’, Econometrica 29(4), 779–800. Goldman, S. M. & Uzawa, H. (1964), ‘A note on separability in demand analysis’, Econometrica 32(3), 387–398. Varian, H. R. (1992), Microeconomic Analysis, 3rd edn, W. W. Norton & Company, New York. 33