A consumer has the following preferences

u(x1,x2)=ax1+bx2

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

(a) Plot this consumer's indifference curves. What is the marginal rate of substitu- tion for this consumer?

(b) Solve the utility maximization problem and write down the Marshallian demand function for good 1 in terms ofp1(takingp2andmas fixed). Also, Plot the demand curve. [Hint: The demand function will display unusual behavior when the price ratio is in the neighborhood ofa/b.]

(c) Supposep1/p2>a/b. What is the cheapest way to obtain a utility level. What if p1/p2

(d) For all subparts below, seta=b= 1. The consumer hasm= 30and facesp1= 1andp2= 2. Plot the budget line, the optimal choice and the indifference curve passing through the optimal choice.

(e) In part (d), supposep1increases top1= 1.5. On the same graph as above, plot the new budget line, the new choice and the indifference curve passing through the new choice. State (in numbers) and depict (on the figure) the income effect and the substitution effect for both goods. (In order to depict substitution and income effects, it might be clearer to label the relevant bundles on the figure and state the income and substitution effects in terms of these label, as opposed to directly labeling them on the diagram.)

(f) Going back to part (d), supposep1had increased top1= 3. On a new graph, once again plot the choice situation in part (d). Also, plot the choice situation corresponding top1= 3. State and depict income and substitution effects.

(g) Use the answer in part (a) to derive the indirect utility function. What is the value of the indirect utility function atm= 30,p1= 4andp2= 2?

(h) Use the answer in part (b) to derive the expenditure function. What is the value of the expenditure function atp1= 4,p2= 2and the utility level found in (g)? What does this tell us about duality?