Reference no: EM132541918

Perfect Substitues A consumer has the following preferences u(x1 , x2) = ax1+ bx2

Suppose the price of good 1 is p1 and the price of good 2 is p2. The consumer has income m.

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

(b) Solve the utility maximization problem and write down the Marshallian demand function for good 1 in terms of p1 (taking p2 and m as fixed). Also, Plot the demand curve. [Hint: The demand function will display unusual behavior when the price ratio is in the neighborhood of a/b .]

(c) Suppose p1/p2 > a/b . What is the cheapest way to obtain a utility level u (hat) . What if p1/p2 < a/b ? What if p1/p2 = a/b ? Follow the clues and write down the consumer's Hicksian demand function for good 1.

(d) For all subparts below, set a = b = 1. The consumer has m = 30 and faces p1 = 1 and p2 = 2. Plot the budget line, the optimal choice and the indifference curve passing through the optimal choice.

(e) In part (d), suppose p1 increases to p'1 = 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 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 your plot and state the income and substitution effects in terms of these label)

(f) Going back to part (d), suppose p1 had increased to p'1 = 3. On a new graph, once again plot the choice situation in part (d). Also, plot the choice situation corresponding to p'1 = 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 at m = 30, p1 = 4 and p2 = 2?

(h) Use the answer in part (b) to derive the expenditure function. What is the value of the expenditure function at p1 = 4, p2 = 2 and the utility level found in (g)? What does this tell us about duality?