More on Indifference Curves and MRS problem 1. A to C

More on Indifference Curves and MRS If you only have two choice variables, you can also find the slope of the level curve and test for diminishing MRS along a given indifference curve, say for k utils, by explicitly solving for x2 as a function of x1 and k and testing whether the second derivative positive. i. Let's try this for the level curve U(X1, X2) = 40 for the utility function U(x1, X2) = x1x2 (i.e., get a formula for x2 as a function of x1, given the condition that xX2 = 40. Differentiate this function to get the slope of the indifference curve at the point (x1, x2) = (2, 10). ii. If you look at the second derivative of the x2 function with respect to x1, - d' x 2 dx ] you can determine the curvature of the level curves of U(X1, X2). If this second derivative is greater than zero, then the indifference curves are consistent with the hypothesis of diminishing MRS. Prove that these indifference curves satisfy diminishing MRS. Now let's do this the way I showed you in class, using the same utility function as part A. i. For U(X1,X2) = X7x2, obtain the slope of the level curve U(x1, X2) = 40 at the point U(X1, X2)= (2, 10) using the formula we derived in class: dx2 = _ U1(X1, X2) dx1 U2 (X1, X2) ii. Using the method I showed you in class, determine if the indifference curves are consistent with the hypothesis of diminishing MRS. The level curves of the function f(X1, X2) = X1X2 in the (X1, X2) plane (where x1 is on the horizontal axis) are all downward sloping. Can you give an explicit formula for a function g(x1X2) whose level curves are also downward sloping, but flatter than the level curves of f(x1, X2)? (In other words, at every point in the plane, g(x1, X2)'s level curve through the point should be flatter than f(x1, X2)'s level curve through that point)