Question

Kate and Antonio meet in their school cafeteria and examine the contents of their lunch boxes....

Kate and Antonio meet in their school cafeteria and examine the contents of their lunch boxes. Kate discovers a bag of M&Ms while Antonio finds a package of Starburst. Suppose a market place for candy has emerged in the school lunch room. The price of a Starburst is 16 cents, p1 = 16, and the price of an M&M is 4 cent, p2 = 4. Antonio has 12 Starbursts and zero M&M’s. Kate has zero Starbursts and 200 M&Ms. Suppose Antonio’s and Kate’s preferences are characterized by marginal rate of substitution functions,
MRSAntonio(x1,x2) = (12)/(√(3(x1))
MRSKate (x1,x2)= (2√(2(x2))/(5)
1. Verify that MRS representation of preferences for Antonio and Kate are consistent with the utility function representations,
uA(x1,x2) = 16√(3(x1)) + 2(x2)
uK(x1,x2) = 2(x1) + 5√(2(x2))
where uA(x) is Antonio’s utility function and uK(x) is Kate’s utility function. (Hint: Use the formula, MRS = MU1/MU2.)
2. Use the formula for optimal demand, MRS (x∗) = p1/p2 together with the equation for the budget line to determine Antonio’s optimal consumption choice. Denote it x∗A.
3. Use the formula for optimal demand, MRS (x∗) = p1/p2 together with the equation for the budget line to determine Kate’s optimal consumption choice. Denote it x∗K.
4. Draw Antonio’s budget constraint the same way as in Problem 3. Illustrate the optimal consumption choice x∗A and the initial allocation eA. Draw the indifference curves that run through each of the two points. Use this to argue how the market has allowed Antonio to improve his welfare relative to his initial allocation.
5. Repeat question 4, but for Kate.

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