Consider a consumer with preferences represented by a utility function u(x1, x2) = x1 + x2. The prices of x1 and x2 are denoted as p1 and p2 and income is denoted as m.

(a) Illustrate the indifference curves of this preference relation.

(b) Write down the Lagrange function for this consumer's problem and derive the first order conditions.

(c) Assuming that the optimal bundle is such that x 1 > 0 and x 2 > 0, under what circumstances are the first order conditions derived above sufficient for an optimum? Show that these conditions are met.

(d) If corner solutions exist, what would be the consumption bundles at these points? State the necessary conditions for each of the two possible corner solutions in terms of the MRS and the slope of the budget line. For this particular utility function, are both of these corner solutions possible? Explain.

(e) Given p = (p1, p2) > 0 and income m > 0, derive the consumer's demand functions.