Consumer Optimisation & Utility Flashcards
(14 cards)
What does a utility function represent?
A utility function assigns a numerical value to each consumption bundle, ranking them according to the consumerβs preferences.
What is a monotonic transformation of a utility function?
Any transformation that preserves the preference ordering. Example: applying a strictly increasing function to a utility function (e.g., multiplying by 2, taking the log).
What is marginal utility?
The additional utility gained from consuming one more unit of a good, holding other goods constant.
How is the MRS related to marginal utility?
MRS XY= MUX/MUY
where πππ is the marginal utility of X and πππ is the marginal utility of Y.
What is the tangency condition in consumer optimisation?
At the optimal consumption bundle:
MRS XY = Px/Py
meaning the rate the consumer is willing to trade goods equals the market rate of trade (the price ratio).
How does the budget constraint factor into optimisation?
The consumer chooses a bundle that satisfies:
PxX+PyY=M
with maximum attainable utility.
What are the steps to mathematically solve for optimal consumption?
Check for convex preferences.
Verify that we have an interior solution.
Set up the tangency condition: equate
MRS and
ππ/ππ
Solve together with the budget constraint to find
πβ and πβ
Why must preferences be convex in the consumer optimisation problem?
Convex preferences ensure diminishing MRS and guarantee that the tangency condition leads to a global maximum (not just a local one).
What is the difference between an interior and a corner solution?
Interior solution: Consumer buys positive amounts of both goods; tangency condition holds.
Corner solution: Consumer buys only one good; tangency condition does not necessarily hold.
What is the Lagrangian setup for the consumerβs problem?
L=U(X,Y)+Ξ»(MβPxXβPyY)
where
π is the Lagrange multiplier (interpreted as the marginal utility of income).
How do you solve the Lagrangian?
1) Take partial derivatives w.r.t.
π, π, and π
2) Set each derivative equal to 0 (first-order conditions).
3) Solve the resulting system of equations.
How does a change in income affect optimal choice?
Normal goods: Consumption rises with income.
Inferior goods: Consumption falls with income.
How does a price change affect choice?
Decompose the effect into substitution and income effects (covered fully in Topic 4).
Why do convexity and monotonicity ensure valid optimisation solutions?
Convexity ensures that βaverages are preferred,β leading to a unique, well-behaved optimum.
Monotonicity ensures that the consumer will spend their full income (budget constraint binds).
