# Essay about Solution Jehle Reny

10807 Words Jun 29th, 2012 44 Pages
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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Let g(t) be an increasing concave (convex) function of a single variable. Show that the composite function, h(x) = g(f (x)) is a concave (convex) function. Answer The composition with an aﬃne function preserves concavity (convexity). Assume that both functions are twice diﬀerentiable. Then the second order partial derivative of the composite function, applying chain rule and product rule, is deﬁned as h (x) = g (f (x)) f (x)2 + g (f (x)) f (x)2

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1 Mathematical Appendix For any concave function, 2 f (x) ≤ 0, the case the two functions are convex: 2 h(x) ≥ 0. g(x) ≤ 0, it should hold 2 h(x) ≤ 0. In 2 f (x) ≥ 0 and 2 g(x) ≥ 0, it should hold
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A1.48 Let f (x1 , x2 ) = −(x1 − 5)2 − (x2 − 5)2 . Prove that f is quasiconcave. Answer Proof: f is concave iﬀ H(x) is negative semideﬁnite and it is strictly concave if the Hessian is negative deﬁnite. H= −2 0 0 −2

2 2 zT H(x)z = −2z1 − 2z2 < 0, for z = (z1 , z2 ) = 0

Alternatively, we can check the leading principal minors of H: H1 (x) = −2 < 0 and H2 (x) = 4 > 0. The determinants of the Hessian alternate in sign beginning with a negative value. Therefore, the function is even strictly concave. Since f is concave, it is also quasiconcave. A1.49 Answer each

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