1-a Say we have a utility function u(x) = xix for a (0, 1). x...

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1-a Say we have a utility function u(x) = xix₂ for a € (0, 1). x = C = R² (a) Construct marginal utilities for both goods. Show the marginal utilities are strictly decreasing. (b) now, construct the Marginal rate of substitution for these preferences. Show the preferences are strictly convex (i.e., the "upper level sets" are strictly convex for every point interior point x E C (c) Now, let u(x) = xix. Show the marginal utilities do not diminish for this consumer (i.e., the utility funciton is not concave). (d) For preferences in part c, show still that preferences are strictly convex. (e) Discuss how the Marginal rate of substitution is related to the slope of an indifference curve at a point x >> 0 (i.e., each component of x is strictly positive) for utility functions given. 1-a Say we have a utility function u(x) = xix₂ for a € (0, 1). x = C = R² (a) Construct marginal utilities for both goods. Show the marginal utilities are strictly decreasing. (b) now, construct the Marginal rate of substitution for these preferences. Show the preferences are strictly convex (i.e., the "upper level sets" are strictly convex for every point interior point x E C (c) Now, let u(x) = xix. Show the marginal utilities do not diminish for this consumer (i.e., the utility funciton is not concave). (d) For preferences in part c, show still that preferences are strictly convex. (e) Discuss how the Marginal rate of substitution is related to the slope of an indifference curve at a point x >> 0 (i.e., each component of x is strictly positive) for utility functions given.