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?