A consumer has demand for two different goods, x and y, which he chooses to maximize the utility function:

U(x, y) = x + y (x 0, y 0)

subject to the budget constraint:

px + qy = m (m, p, q > 0)

where p is the price of good x, q is the price of good y, and m is total income

a) Show that the utility function is concave

b) Write out the Lagrangian for the constrained maximization problem, assuming consumer spends his full income

c) Find the utility maximizing demands for both goods, as well as the Lagrange multiplier, all as functions of the three exogenous variables

d) Are the demand functions which you have found homogeneous in prices and income (prove your answer) ? If yes, what is the degree of homogeneity?

Show all your work.