Solution

1. Let L be Larry's hours of work. His use of time is constrained by the following equation:

   L + y = 12

   Working L hours generates w*L dollars of income that he can use to buy beer. His money constraint is thus:

   w*L = P*x

   Since he only cares about y and x, it is natural to use the second equation to eliminate L from the first:

   L = (P/w)*x

   (P/w)*x + y = 12

2. Finding the equilibrium value of x: x = 3*w/P = 3*5/1 = 15 cans. The graph is shown below:

   

3. The increase in Larry's wage causes his budget constraint to rotate outward from BC1 to BC2: he is able to consume more beer for any given amount of leisure. His new consumption of beer is x = 3*6/1 = 18 cans and the graph looks as follows: