A consumer has the following preferences

u(x1, x2) = log (x1) +x2

Suppose the price of good 1 isp1and the price of good 2 isp2. The consumer has incomem.

(a) Find the optimal choices for the utility maximization problem in terms ofp1, p2 andm.Denote the Lagrange multiplier by.

(b) How do the optimal choices change asmincreases? What does the income offer curve (also called the income expansion path) look like for this consumer? (You can show it on a diagram.)

(c) What is the slope of the Marshallian demand curve for good 1? Use the Slutsky equation to find the slope of the Hicksian demand curve for good 1, without actually solving the expenditure minimization problem.

(d) For a utility level, solve the expenditure minimization problem and find the optimal choices in terms ofp1,p2and.Denote the Lagrange multiplier by.

(e) Find the Hicksian demand curve for good 1. What is the slope of this curve? Does it match your answer in (c)?

(f) Find the expenditure function. Find its partial derivative with respect top2. Provide an interpretation of this derivative in terms of choice behavior.

(g) Use the answer in (e) to find compensating variation for a change inp1froma tob,a < b. [Hint: d/dxlog(x) =1/x

and the fundamental theorem of calculus tell us

b

cf'(t)dt=c(f(b)f(a))

a

wherecis a constant.]

(h) Use the answer in (f) to find compensating variation for a change inp1froma

tob. Does this match your answer in(g)?

(i) What is the consumer surplus lost as a result of the change inp1fromatob,

a < b?