The utility function is u(x, y) = x + y The price of good x is px and the price of good y is py. We denote income by M > 0. This function is well-defi ned for x > 0 and for y > 0. so, assume x > 0 and y > 0 unless otherwise stated.

1. Is the utility function increasing in x? Is the utility function concave in x?

2. Write down the maximization problem of the consumer and explain why the budget constraint is satisfied with equality.

3. Write down the Langrangian function.

4. Write down the first order conditions for this problem with respect to x, y, and .

5. Solve explicitly for x and y as a function of px, py, and M.

6. Are these points maxima of the problem above? Check that the determinant of the bordered Hessian is positive at x , y , and .

7. Can you guess the solutions for x and y for the following maximization problem?