I have two 5-year old girls — Ellie and Jenny — at home. Suppose I begin the day by giving each girl 10 toy cars and 10 princess toys. I then ask them to plot their indifference curves that contain these endowment bundles on a graph with cars on the horizontal and princess toys on the vertical axis.
A. Ellie’s indifference curve appears to have a marginal rate of substitution of −1 at her endowment bundle, while Jenny’s appears to have a marginal rate of substitution of −2 at the same bundle.
(a) Can you propose a trade that would make both girls better off?
(b) Suppose the girls cannot figure out a trade on their own. So I open a store where they can buy and sell any toy for $1. Illustrate the budget constraint for each girl.
(c) Will either of the girls shop at my store? If so, what will they buy?
(d) Suppose I do not actually have any toys in my store and simply want my store to help the girls make trades among themselves. Suppose I fix the price at which princess toys are bought and sold to $1. Without being specific about what the price of toy cars would have to be, illustrate, using final indifference curves for both girls on the same graph, a situation where the prices in my store result in an efficient allocation of toys.
(e) What values might the price for toy cars take to achieve the efficient trades you described in your answer to (d)?
B. Now suppose that my girls’ tastes could be described by the utility function u(x1,x2) = xα 1 x2(1−α) , where x1 represents toy cars, x2 represents princess toys and 0< α < 1.
(a) What must be the value of α for Ellie (given the information in part A)? What must the value be for Jenny?
(b)When I set all toy prices to $1, what exactly will Ellie do? What will Jenny do?
(c) Given that I am fixing the price of princess toys at $1, do I have to raise or lower the price of car toys in order for me to operate a store in which I don’t keep inventory but simply facilitate trades between the girls?
(d) Suppose I raise the price of car toys to $1.40, and assume that it is possible to sell fractions of toys. Have I found a set of prices that allow me to keep no inventory?