Essential Problems in bold: Demand with a quasi-linear utility function

Consider a consumer with the following utility function u(x1, x2) = √x1 + x2. This is an example of a Quasi-Linear utility function. We can see that it is linear in good 2 but not in good 1.

(a) In a maximum of two lines define with words what is the Marginal Rate of Substitution (MRS).

(b) Calculate the MRS for this consumer for general values of x1 and x2. (c) Explain in words what is the role of the MRS = p1/p2 condition in consumer’s theory.

(d) Let’s assume that m = 10 and p1 = 2 and p2 = 8. Which bundles simultaneously satisfy that MRS = p1/p2 and that total expenditure equals m?

(e) Which bundles simultaneously satisfy that MRS = p1/p2 and that total expenditure equals m if p1 goes down to 1? Is that possible?

(f) What will be the consumer’s optimal choice in this last case? What is the MRS for that optimal bundle? (The latter may be an ugly number).

(g) Based on the previous points, derive the demand functions for both goods: x∗1(p1, p2, m) and x∗2(p1, p2, m).