Carlos's utility function is

u$($a$,$b$)=$a$+2$b$,$

where a is his consumption of artichokes and b is his consumption of broccoli.

Laura's utility function is

u$($a$,$b$)=$min$\{$a$,2$b$\}.$

Laura initially has $60$artichokes and no broccoli. Carlos initially has $60$broccoli and no artichokes Let ac be Carlos's consumption of artichokes and let bc be his consumption of broccoli. Let artichokes be the numeraire, and let the price of broccoli be p$.$At competitive equilibrium, Carlos's consumption will have to satisfy the budget constraint

ac $+\_\_$bc $=\_\_$p

This is one equation in two unknowns. To find a second equation, consider Laura's consumption. In competitive equilibrium, total consumption of artichokes equals the total supply of artichokes, and total consumption of broccoli equals the total supply of broccoli. Therefore, Laura will consume $60$ac artichokes, and she will consume $60$ bc broccoli.

At a competitive equilibrium, Laura will be consuming at one of her kink points. The kinks occur at bundles where she consumes $2$ artichokes for every broccoli she consumes. Therefore, we know that $60-$ac $/60-$bc$=2.$

You can solve the two equations that you found in Parts $1$ and $2$ to find the quantities of artichokes and broccoli consumed in competitive equilibrium by Carlos and Laura. Carlos will consume $\_\_\_$ artichokes and $\_\_\_\_$ broccoli. Laura will consume $\_\_\_$ artichokes and $15$ broccoli.