Problem 4 - Trade and MRS (40 points) Antonio and Kate have preferences over Starbursts and M&Ms (goods 1 and 2, respectively) that are represented by utility functions, UA (X1, 12) = 2VT1 + 12 UK (T1, [2) = 2VT1+ 12 where uA is Antonio's utility function and ux is Kate's utility function. Antonio has an initial endowment of eA = (16, 0) and Kate's initial endowment bundle is ex = (0, 6). 1. Determine Kate and Antonio's respective utility levels if they are to simply consume their initial endowments, up(ex) and uA(eA). 2. Determine MRSA (r), that is Antonio's marginal rate of substitution given consumption bun- dle r. Determine MRSA(eA). 3. Determine MRSk (x), that is Kate's marginal rate of substitution given consumption bundle r. Consider a hypothetical endowment bundle ex = (, B). Determine Kate's marginal rate of substitution in her endowment bundle as MRSk (ex) = lime-0 MRSk (ex) - It is ok to state a non-existent limit that increasing or decreasing without bound as either too or -co. 4. Determine the equation for Antonio's indifference curve that runs through his endowment bundle, ex. Draw it. 5. Let us consider if by engaging in trade, Antonio and Kate can improve their outcomes relative to simply consuming their respective endowment bundles. Specifically, taking their endow- ments as their initial conditions, consider a trade where Kate trades with Antonio to receive Starbursts in return for M&M's. To begin with, consider a trade where Kate acquires 4 Starbursts from Antonio. 2 (a) What is the least number of M&M's Antonio will require to be willing to make the trade? For these exercises, assume candy is perfectly divisible. (b) Would Kate be willing to pay this amount of M&M's? What is the most Kate would have been willing to pay? 6. Based on your insights from questions 5, I want to you to work toward an understanding of how the insights relate to Antonio and Kate's marginal rates of substitution. So, instead of Kate trading for 4 Starbursts, specifically, let's make the trade for da units of Starbursts. Denote by d'ar2 the least units of M&M's that Antonio would require in exchange for dri Starbursts. (a) Determine an expression for dar2. (b) Calculate Antonio's exchange rate dar2/dr, for dri = 1 and daj = .1, respectively. (c) As you consider the limit daj - 0, compare dAr2/dr] to Antonio's marginal rate of substitution MRS(eA). Comment. 7. Now, let us consider the thought experiment that Kate makes a series of trades. Each time, Kate trades for dr, = 2 units of Starburst in exchange for the least number of M&M's Antonio is willing to accept to make the trade given Antonio's current candy bundle at the time of the trade. Kate carries out a trade if she is willing to pay this amount. That is, the trade only takes place if both parties agree. Each trade changes both Antonio's and Kate's candy bundles. (a) For each trade, determine d'ar2 given Antonio's current candy bundle. Verify that Kate is willing to make the trade. Write up the series of trades and record how Kate and Antonio's candy bundles are changing. In each "round" also state Kate's and Antonio's marginal rate of substitutions at their respective current candy bundles. Illustrate An- tonio's resulting candy bundle after each successive trade in the same figure you drew Antonio's indifference curve. (b) At what point does the series of trades end? Be sure to State Kate's and Antonio's MRS at their final candy bundles. (c) Consider Kate's end-of-trades candy bundle. i. How many Starbursts has Kate acquired? ii. In total, how many M&M's did Kate end up paying for them? iii. How many M&M's would Kate have been willing to pay for them. iv. The difference between what Kate would have been willing to pay and what Kate actually paid is a measure of her surplus from the trade. State Kate's trade surplus