Problems

$5\mathrm{.}1$

Thirsty Ed drinks only pure spring water, but he can purchase it in two different$-$sized containers: $0\mathrm{.}75$ liter and $2$ liter. Because the water itself is identical, he regards these two "goods" as perfect substitutes.

a$.$ Assuming Ed's utility depends only on the quantity of water consumed and that the containers themselves yield no utility, express this utility function in terms of quantities of $0\mathrm{.}75-$liter containers $(x)$ and $2-$liter containers $(y).$

b$.$ State Ed's demand function for $x$ in terms of $p_x{p}_y,$ and I.

c$.$ Graph the demand curve for $x,$ holding I and $p_y$ constant.

d$.$ How do changes in I and $p_y$ shift the demand curve for $x?$

e$.$ What would the compensated demand curve for $x$ look like in this situation?

$5\mathrm{.}2$

David N$.$ gets $$3$ per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter $($at $$0\mathrm{.}05$ per ounce$)$ and jelly $($at $$0\mathrm{.}10$ per ounce$).$ Bread is provided free of charge by a concerned neighbor. David is a particular eater and makes his sandwiches with exactly $1$ ounce of jelly and $2$ ounces of peanut butter. He is set in his ways and will never change these proportions.

a$.$ How much peanut butter and jelly will David buy with his $$3$ allowance in a week?

b$.$ Suppose the price of jelly were to increase to $$0\mathrm{.}15$ an ounce. How much of each commodity would be bought?

c$.$ By how much should David's allowance be increased to compensate for the increase in the price of jelly in part $($b$)?$

d$.$ Graph your results in parts $($a$)$ to $($c$).$

e$.$ In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity.

f$.$ Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly.