Unformatted text preview: 14.04 - Problem Set 1 Due Sept 22nd in recitation 1) Start with an arbitrary utility function ux 1 , x 2 that is differentiable. Let vu be a monotonic transformation of u. a) Solve: max ux 1 , x 2 ST : p 1 x 1 p 2 x 2 m b) Solve: max vux 1 , x 2 ST : p 1 x 1 p 2 x 2 m c) Discuss the relationship between these problems. What characteristics of the utility function is generating this result? 2) Consider the following problem: max x y x,y Subject to: x py 10, x 0, y 0 a) Show formally that the utility function x y is at least weakly monotonic and strongly convex for 0. You may use ideas from problem 1 to simplify the problem. b) Find V, p, x, p, y, p 3) Solve the following: max ln x y x,y ST : 2x y 10, x 0, y 0 4) One way to rule out the potential that the non negativity constraints aren't binding is to look at the marginal rate of substitution (MRS) when one of the factors gets arbitrarily close to zero. Suppose that we have a function fx 1 , x 2 . The MRS 12 x 1 , x 2 is the amount of x 1 required to keep the function f the same when x 2 changes by a small amount. MRS 12 x 1 , x 2 is read "the marginal rate of substitution of good 1 for good 2 at x 1 , x 2 " Formally: MRS 12 x 1 , x 2 dx 1 dx 2 x 1 ,x 2 fx 1 ,x 2 x 2 fx 1 ,x 2 x 1 a) Consider the function fx, y xy. Starting from a point where x, y 0, what happens to the MRS xy as y grows smaller and approaches zero (ie lim y0 MRS xy x, y) ? What happens to lim x0 MRS yx ? b) Consider the function x y. What is lim y0 MRS xy ? What is lim x0 MRS yx ? c) Consider ln x y. What is lim y0 MRS xy ? What is lim x0 MRS yx ? ...
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This homework help was uploaded on 02/01/2008 for the course ECON 14.04 taught by Professor Izmalkov during the Fall '06 term at MIT.
- Fall '06