Choice & Demand

Question 1

How would you find the optimum choice for a consumer with income m and utility function u(x, y) = xy facing prices p, q ?

Answer 1

Since the preferences represented by the utility function are well behaved, there is an interior solution
Rearrange the budget equation to get one good on its own, substitute that into the utility function, and then use the FOC to find the optimum
Equivalently, the MRS (slope of IC) and price ratio (slope of BL) can be equated (tangency condition)

Question 2

When can the tangency condition be used to find a consumer’s optimal choice within their budget set?

Answer 2

If preferences are convex, monotonic, and there is a point where the tangency condition holds, it is definitely an optimum
For corner solutions, substituting BL into u can still work

Question 3

Give an example of when strictly convex preferences do not yield the optimum through the tangency condition?

Answer 3

A quasilinear utility function u(x, y) = ln(x) + y could have a corner solution if the BL is flatter than the IC at any point

Question 4

How can the tangency condition be rewritten to understand it better?

Answer 4

The tangency condition is that MRS = -p/q
Equivalently MU₁/p = MU₂/q
If the first ratio was larger than the second the consumer would sell y and buy x (good x “better value”)
Similarly |MRS| > p/q means good x is a better deal

Question 5

Why does the tangency condition fail at corner solutions?

Answer 5

Since the MRS and price ratio are not equal the consumer would prefer to sell some of one good to buy the other but has none of that good left to sell

Question 6

Which of the assumptions for well-behaved preferences ensures that solutions lie on the budget line?

Answer 6

Monotonicity

Question 7

Which of the assumptions for well-behaved preferences rules out multiplicity of optimal bundles in a standard budget set and satisfaction of the tangency condition at sub-optimal bundles?

Answer 7

Strict convexity

Question 8

How is the demand function denoted for well-behaved preferences?

Answer 8

x*(p, m) = most preferred bundle in B(p, m)

Question 9

What is the Cobb-Douglas demand function?

Answer 9

From u(x₁, x₂, …) = x₁^α₁x₂^α₂… and corresponding budget constraint, can find that x_i* = α_im/p_i using condition that MU_i/p_i must be equal for all goods

Question 10

What can be said about finding demand functions more generally?

Answer 10

The optimal bundle x* must have equal MU_i/p_i for all i with x_i* > 0 (found by inspection)
Can then solve the system of equations from marginal utilities and budget equation

Question 11

How do income effects characterise goods?

Answer 11

Demand for normal goods rises as income rises (all goods for Cobb-Douglas utility function)
Demand for inferior goods falls as income rises (this can’t apply at all income levels otherwise it would be a bad not a good)

Question 12

How do own price effects characterise goods?

Answer 12

Demand for ordinary goods rises as the price of the good falls
Demand for Giffen goods falls as the price of the good falls

Question 13

What is the Hicks decomposition of the effect of a change in a price?

Answer 13

The substitution effect shifts along the same IC to the point tangent to a budget line under the new price ratio, then the income effect shifts to a new IC
This decomposition comes from the idea that a pure substitution shouldn’t change the consumer’s welfare

Question 14

What is compensated demand and how does it relate to the substitution effect?

Answer 14

The compensated demand is the consumer’s new choice under the new prices but with an income compensated to keep the consumer on the same IC
The difference between original demand and compensated demand is the substitution effect

Question 15

What is the Slutsky decomposition of the effect of a change in a price?

Answer 15

Pivot the BL around the original choice so it matches the new price ratio, the substitution effect shifts up to the optimum on this BL, then the income shifts to the optimum when the BL is shifted to its actual level

Question 16

What are the signs of the substitution and income effects?

Answer 16

The substitution effect is always in the opposite direction of the price change
For normal goods the income effect is in the opposite direction of the price change, for inferior goods the same direction

Question 17

How can Giffen goods be explained/characterised using the substitution and income effects?

Answer 17

The magnitude of the substitution effect must be smaller than the income effect (inferior good) so there are no close substitutes and a large fraction of income is spent on the good

Question 18

How do cross-price effects characterise goods?

Answer 18

The demand for a substitute rises when the price of the other good rises
The demand for a complement falls when the price of the other good rises
Not every good has a cross-price effect and good 1 being a substitute/complement for good 2 does not imply that good 2 is a substitute/complement for good 1