Max is a utility-maximizer consumer with a utility function U (x , y )=x+ y and the usual budget

constraint M=px x+ p y y , where M is his income, px and py is the price of good x and y,

respectively.

1.

Write down Max's optimization problem as an optimization problem with a single variable, y.

(Hint: solve the budget constraint for x, and plug the solution for x into the objective function.

2.

Write down Max's first order condition(s) for an interior solution.

3.

Use the implicit functions theorem with the first order condition to calculate all the partial

derivatives of the optimal value of y. Note: you have to calculate these derivatives without

calculating the optimal value of y!

4.

Find the optimal value of y and use it to verify your results from question 3.

5.

Write down the second order condition(s) and check whether it/they are satisfied.

Of course Max cannot consume a negative value of y, and he also cannot consume more than the

maximum possible quantity of y, M/py.

6.

Is is possible Max has a corner solution y = 0 ? How about a corner solution y = M/py ? If yes,

for which values of prices and income are they possible?