Consumption vs Leisure

Larry Laidback likes to lie in the sun drinking beer and working on his tan. His preferences can be represented by a utility function of the form:

U = x^0.25 * y^0.75

where y is the number of hours he spends in the sun and x is the number of cans of beer he drinks. There's only one catch: if he wants to drink any beer, Larry must spend part of each day sober working in a dark office.

1. Suppose Larry has 12 hours a day which he can either spend lying in the sun or going to work. If he works, he gets paid w dollars per hour. On the other hand, each beer he drinks costs P dollars. Show that his budget constraint can be written as:

   12 = (P/w) * x + y

2. Given Larry's utility function, it can be shown that his demand equations for cans of beer and hours in the sun will be the following: x = 3w/P and y = 9. Using this information, graph his equilibrium when the price of beer is $1 per can and his wage is $5 per hour. Show his budget constraint, his indifference curve and his optimum consumption bundle. Label the intercepts of the budget constraint and calculate Larry's optimum consumption of hours in the sun and cans of beer.
3. Suppose Larry gets a raise to $6 per hour. Graph his new equilibrium.