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please just answer 3 and 4. It is very urgent. I will give you thumb up. It's an abstract mathematics

it's an applied mathematics in econometrics

I Preference and Choice 1. Suppose that f:R R is a strictly increasing, differentiable function and u:X + R is a differentiable utility function representing the preference relation Prove that, if y(x) = f(u(x)), then maximization of u() or v(-) will yield the same preference-maximizing behaviour * EC(BS). 2. Suppose that XER. Consider the preference relation given by: * Ey > or x = n and x where x = (x,x) E X and y (41,42) E X. Characterize the indifference sets for this preference relation. Can these preferences be represented by a utility function? Explain your answer. 3. Suppose that X = {w, 1,4,7). Consider the choice structure denoted by (9,C()), such that B = {{w.y, },{x,y}, {y, z}, {x,}, {x,y,z}} and: C({w.y =)) = {w} C({x,y})= {2} C({1,2}) = {v} C(2, 2)) = {2}: What choice rules for C({r.y, 2}) satisfy WARP, if any? Prove your answer. II Consumer Choice 4. Suppose that the Walrasian demand function x(p.u) satisfies WARP. Prove that (ap, ow) = rp, w) for any a > 0. 5. Solve exercise 2.E.2 in Mas-Colell, Whinston and Green and briefly discuss the intuition behind these elasticity formulas. 6. In the Consumer Choice lecture notes, we established that if a Walrasian demand function x(p,w) satisfies the following properties: Homogeneous of degree zero Walras' law WARP then it follows that, if the price of commodity & decreases, its compensated demand 1() must necessarily increase. Suppose X R. What does this result imply for the compensated cross-price effect? Prove your answer. 2 3. Suppose that X = {w,,y,). Consider the choice structure denoted by (98,C(-)), such that B = {{w,y,z}, {r,y}, {y, z},{x,z},{r,y,z}} and: C({w.y =)) = {w} C({x,y}) = {u} C({y,z)) = {v} C({.,2)) = {2} What choice rules for C({6,7,2}) satisfy WARP, if any? Prove your answer. II Consumer Choice 4. Suppose that the Walrasian demand function x(p.w) satisfies WARP. Prove that rap, aw) = (p, w) for any a > 0. I Preference and Choice 1. Suppose that f:R R is a strictly increasing, differentiable function and u:X + R is a differentiable utility function representing the preference relation Prove that, if y(x) = f(u(x)), then maximization of u() or v(-) will yield the same preference-maximizing behaviour * EC(BS). 2. Suppose that XER. Consider the preference relation given by: * Ey > or x = n and x where x = (x,x) E X and y (41,42) E X. Characterize the indifference sets for this preference relation. Can these preferences be represented by a utility function? Explain your answer. 3. Suppose that X = {w, 1,4,7). Consider the choice structure denoted by (9,C()), such that B = {{w.y, },{x,y}, {y, z}, {x,}, {x,y,z}} and: C({w.y =)) = {w} C({x,y})= {2} C({1,2}) = {v} C(2, 2)) = {2}: What choice rules for C({r.y, 2}) satisfy WARP, if any? Prove your answer. II Consumer Choice 4. Suppose that the Walrasian demand function x(p.u) satisfies WARP. Prove that (ap, ow) = rp, w) for any a > 0. 5. Solve exercise 2.E.2 in Mas-Colell, Whinston and Green and briefly discuss the intuition behind these elasticity formulas. 6. In the Consumer Choice lecture notes, we established that if a Walrasian demand function x(p,w) satisfies the following properties: Homogeneous of degree zero Walras' law WARP then it follows that, if the price of commodity & decreases, its compensated demand 1() must necessarily increase. Suppose X R. What does this result imply for the compensated cross-price effect? Prove your answer. 2 3. Suppose that X = {w,,y,). Consider the choice structure denoted by (98,C(-)), such that B = {{w,y,z}, {r,y}, {y, z},{x,z},{r,y,z}} and: C({w.y =)) = {w} C({x,y}) = {u} C({y,z)) = {v} C({.,2)) = {2} What choice rules for C({6,7,2}) satisfy WARP, if any? Prove your answer. II Consumer Choice 4. Suppose that the Walrasian demand function x(p.w) satisfies WARP. Prove that rap, aw) = (p, w) for any a > 0