$[25$ points$]$ Chuck, who thinks he has a condition in which he is extremely sensitive to light and

electromagnetic currents, believes that his condition will be eased by drinking smoothies made up of Fuji

apples and berries, mixed in just the right proportions. Suppose that he has M $=$ $$90$ to spend on apples

and berries and their prices are PA $=$ $$3$ per pound and PB $=$ $$3$ per pound, and that his utility function is

U$($A$,$ B$)=$ min$(3$A$,2$B$)$

where A is pounds of apples and B is pounds of berries. The real function F$($X$,$ Y$)=$ min$($X$,$ Y$)$ outputs the

smaller of X and Y$,$ or specifically min$($X$,$ Y$)=$ X if X $<=$ Y and min$($X$,$ Y$)=$ Y if X $>$ Y$.$ Thus, the above

utility function compares $3$A and $2$B and outputs the smaller of the two.

a$)$ Neatly plot an indifference curve through $($A$,$ B$)=(30,30).$

b$)$ Draw the budget line.

c$)$ Write out the two equations that characterize the solution to Chuck$$s utility maximization problem. One

equation is the budget line. The other equation characterizes the proportions of apples and berries that

Chuck prefers, or geometrically the corners of all of the L$-$shaped indifference curves.

d$)$ Give Chuck$$s optimal consumption bundle $($A$*,$ B$*).$

e$)$ Show $($A$*,$ B$*)$ on your graph along with the indifference curve through $($A$*,$ B$*).$