Question

Suppose Humphrey and Matilda live together. Humphrey currently smokes 20 packs of cigarettes per month; Matila hates the smoke. They currently have no agreement restricting smoking. Their only going expense is monthly rent, which they split 50:50. Draw an Edgeworth box with two goods - smoke and rental payments. Make up some reasonable indifference curves. Show the initial endowment. What Pareto efficient points might result from bagaining to restrict smoke? How does the graph show what price per pack Matilda might pay to buy down Humphrey's smoking (i.e., show the relative prices on your figure)? How would your answers change if the status quo is that the two have an agreement for no smoking and Humphrey would like to smoke as much as 20 packs per month? He must seek Matilda's permission to do so. (Hint: For Matilda, redefine Humphrey's smoking as smoke reduction.)

Answer 1

We need to draw an edgeworth box showing the changes in utility of both Humphrey and Matilda with change in the consumption of smoke and rent paid.

Humphrey's utility increases as he smokes but it is a bad for Matilda . Humphrey's utility depends on packs of cigarettes ie a good and rental payments ie a bad. He therefore has upward sloping indifference curves with one good and one bad . however Matilda's utility depends on her rent payments and Humphrey 's smoke both of which are bad for her . only when he reduces the smoke consumption , it becomes a good for her .

The top right corner is the starting point of Matilda and the bottom left for Humphrey . as we see , the reduction in smoke increases the utility for Matilda and it is opposite for Humphrey . also we have 2 extreme points , which are the best possible points for either of them . as we see in the Edgeworth box the top left corner is the point where Matilda has the best possible point because there she pays no rent ahs there is no smoke which is the least desirable for Humphrey . similarly at C we have the most desirable for Humphrey and least fro matilda.

now we look at the second edgeworth box.

in the second edgeworth box it is seen that at point B both pay equal ie 50-50 rent as in the question and given in question is Humphrey smokes 20 cigarettes a month.  initially with just rent agreement and no smoke agreement , the indifference curve ICs of both of them didnt meet anywhere in the edgeworth box so there is no pareto optimal point .At E2, their indifference curves are tangent, and no further mutually beneficial trades are possible. we reach here by possible bargaining . since ICs of both the persons form a tangent at E2 , Matilda would have this agreement for reduction in smoking .

now we come to the other part of edgeworth box where the rent is split equally but no smoking agreement which is point B . so in absence of an agreement , Humphrey would pay for the right to smoke. hence we get to point E1 where he buy the rights to smoke from his roommate. At E2 its is mre better for Humphrey but at E1 for matilda though both are pareto optimal points.