CONSUMPTION. (Lectures 4, 5, and 6) Remark: (*) signals those exercises that I consider to be the most important

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1 CONSUMPTION (Lectures 4, 5, and 6) Remark: (*) signals those eercises that I consider to be the most imortant Eercise 0 (MWG, E. 1.B.1, 1.B.) Show that if is rational, then: 1. if y z, then z;. is both irrefleive ( never holds) and transitive (if y and y z, then z);. ~ is refleive ( ~ for all ), transitive (if ~y and y~z, then ~z), and symmetric (if ~y, then y~ ). Eercise 1 Prove that strong monotonicity imlies local nonsatiation, but not vice versa. Eercise Assume that there are only two goods in one economy. Draw indifference curves that (a) satisfy and (b) violate each of the following roerties: 1. transitivity;. strict conveity;. conveity; 4. monotonicity. Eercise Show that if there eists a utility function that reresents, then must be rational.

2 Eercise 4 (MWG, E..C.1) Assume that there eist only two goods, good 1 and good. Define y if either 1 > y 1 or 1 = y 1 and y. This is known as the leicograhic reference relation. Verify that the leicograhic ordering is comlete, transitive, strongly monotone, and strictly conve. Eercise 5 Show that leicograhic references are not continuous. Eercise 6 Show that the eenditure function e(,u) satisfies the following roerties: 1. non increasing in i, i=1, n;. homogeneous of degree 1 in ;. concave in ; 4. continuous in, 0. Eercise 7 (*) Let u( 1, )= k a 1-a 1, for 0<a<1. 1. Solve the utility maimization roblem and find the demand functions;. Verify that i (,m), i=1,, satisfies homogeneity of degree 0 in (,m) and Walras law. Eercise 8 (*) a a 1/a Consider the CES utility function u()= ( 1 + ), for a Comute the demand functions;. Derive the indirect utility function;. Derive the demand corresondences and indirect utility functions for the linear utility and the Leontief utility cases. Show that the functions comuted in 1. and. aroach these as a 1 and as a -, resectively;

3 4. Comute the Hicksian demand functions and verify their roerties. Eercise 9 (*) Let (-, ) L-1 + denote the consumtion set and let the utility function be u()= 1+ w(,,.. L ). 1. Show that the demand functions for goods,,l are indeendent of income;. Argue that the indirect utility function can be writtten in the form v(,m)= m+φ() for some function φ( ); Now, for simlicity, let L= and suose Are the demand functions still indeendent of income? 4. For a given fied level of rices, eamine how demand changes as income increases. Eercise 0 (from MWG, E..D.6) Consider the three-good setting in which the consumer has utility function u()= ( 1 - b 1 ) a ( - b ) c ( - b ) d. 1. Why can you assume that a+c+d=1 without loss of generality? Do so for the rest of the roblem.. Write down the first-order conditions for the UMP, and derive the consumer s Walrasian demand and indirect utility functions. This system of demand is known as the linear eenditure system and it is due to Stone (1954);. Verify that these demand functions satisfy the following roerties: homogeneity of degree 0 in (,m), Walras Law, conveity/uniqueness. Eercise 1 V, E. 7.,. 114 Eercise V, E. 7.4,

4 Eercise V, E. 7.6, arts a) and b),. 115 Eercise 4 (from MWG, E..G.) Consider the linear eenditure system utility function given in Eercise Derive the Hicksian demand and eenditure functions. Check their roerties;. Verify that the Slutsky equation holds;. Verify that the own-substitution effects are negative and that the comensated cross-rice effects are symmetric. Eercise 5 The consumer buys bundles i at rices i, i=0,1. Justify whether each of the following choices staisfies the weak aiom of revealed reference: 1. 0 =(1,), 0 =(4,), 1 =(,5), 1 =(,1);. 0 =(1,6), 0 =(10,5), 1 =(,5), 1 =(8,4);. 0 =(1,), 0 =(,1), 1 =(,), 1 =(1,); 4. 0 =(,6), 0 =(0,10), 1 =(,5), 1 =(18,4). Eercise 6 (MWG, E..F.16) (*) Consider a setting where L= and a consumer whose consumtion set is. Suose that his demand function (, is 1 (, =, 1 (, =, w (, =. 1. Show that (, is homogeneous of degree 0 in (, and satisfies Walras law;. Show that (, violates the weak aiom;. Show that v. S(,.v = 0 for all vœ. Eercise 7 (MWG, E..F.17) In an L-commodity world, a consumer s Walrasian demand function is w k (, = L for k=1,,l. l = 1 l 5

5 1. Is this demand function homogeneous of degree 0 in (,?. Does it satisfy Walras law?. Does it satisfy the weak aiom? Eercise 8 V, E. 8.,. 140 Eercise 9 V, E. 8.5,. 141 Eercise 40 V, E. 8.6,. 141 Eercise 41 V, E. 8.7,. 141 Eercise 4 V, E. 8.10,. 141 Eercise 4 V, E. 8.1,. 14 Eercise 44 V, E. 8.16,. 14 Eercise 45 V, E. 9.10,. 159 Eercise 46 V, E. 9.11,. 159 Eercise 47 V, E. 10.,

Notes I Classical Demand Theory: Review of Important Concepts

Notes I Classical Demand Theory: Review of Important Concepts The notes for our course are based on: Mas-Colell, A., M.D. Whinston and J.R. Green (1995), Microeconomic Theory, New York and Oxford: Oxford