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ECON5101 price vector

创建日期：2024-05-30 17:17:48

所属分类：经济Economics

标签： ECON5101

price vector

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ECON5101

Motivation
Elements of a decision problem:
What is feasible?
What is desirable?
We can now put them together to talk about choice. We will:
1 solve the consumer’s optimisation problem to derive their individual demand function. (now)
2 look at some comparative statics: how consumer demand changes as exogenous variables change.
(next)

Assumptions about Individuals and Preferences
Behavioural postulate: An individual always chooses their most preferred alternative among available
choices.
In a market environment where the consumption space consists of two perfectly divisible goods
(X = R2+), and the price vector p = (p1, p2) is given,
their feasible set is B(p,m) = {x ∈ R2+ : p1x1 + p2x2 ≤ m},
their preferences are described by the rational preference relation ⪰,
they will choose a bundle that is both affordable and weakly preferred to all other affordable
bundles.
Assume that preferences are continuous.

Rational Choice: Graphically
For now, also assume strictly convex and
strongly monotone preferences.
We will choose the bundle that lies on the
highest indifference curve we can ”reach.”

Rational Choice: Graphically
At the optimal bundle (x∗1 , x
∗
2 ),
the indifference curve it belongs to is tangent
to the budget constraint,
the budget is fully exhausted.
This rests on the assumptions we made on the
previous slide, and won’t always be true otherwise.

Rational Choice
At the optimal bundle (x∗1 , x
∗
2 ),
the indifference curve it belongs to is tangent to the budget constraint,
the budget is fully exhausted.
We will now see how to derive optimal choice more formally, as the solution to a constrained
optimisation problem. We will:
discuss the assumptions we need for the two above conditions to characterise the optimum.
discuss how to derive optimal choice in those circumstances.
do some examples, including those in which our assumptions don’t hold.

1 Constrained Utility Maximisation
Assumptions
Individual Demand
Cobb-Douglas Utility Functions
2 Optimisation with Other Common Utility Functions
Violation of Assumptions
Quasilinear Utility Functions
Perfect Substitutes
Perfect Complements

Constrained Utility Maximisation
We need a systematic approach to the consumer’s optimisation problem. Instead of the rational
preference relation ⪰, we will work with a utility function that represents it.
Utility Maximisation Problem (UMP)
Given price p = (p1, p2) with p1, p2 > 0, a consumer with income m > 0 solves the following problem:
max
x1,x2≥0
u(x1, x2) subject to p1x1 + p2x2 ≤ m.
Assumptions: The utility function u : R2+ → R is continuous, differentiable, strictly increasing, and
strictly concave.
Notation: For convenience, I will occasionally write (x1, x2) using the vector boldface, x.

Assumptions
u is continuous
Ensures the existence of a solution to the UMP.
Since we assumed our preferences are rational and continuous, the Utility Representation Theorem
tells us that we will be able to represent them with a continuous utility function.
u is strictly increasing
Ensures the consumer spends all their income (Walras’ Law), i.e., p1x
∗
1 + p2x
∗
2 = m.
This is guaranteed when preferences are strongly monotone.
u is strictly concave
Ensures a unique solution the UMP.
This is guaranteed when preferences are strictly convex.

Assumptions
Claim: If u is strictly concave, then there is a unique solution to the UMP.
Proof (not assessed):

Assumptions
Claim: If preferences are strictly convex, then a utility function u which represents these preferences is
strictly concave..
Proof (not assessed):

Individual Demand
Definition 1: Demand
For each price-income combination, the consumer solves the UMP to obtain an optimal consumption
bundle. The function that relates the optimal quantity of good i to prices and income is called the
demand for good i , xi (p,m).

Deriving Individual Demand
The Lagrangian for the UMP, given prices p and income m > 0 is
L = u(x)− λ(p1x1 + p2x2 −m).
Our assumptions that u is strictly concave, strictly increasing, and continuously differentiable ensure
that first order conditions are necessary and sufficient for constrained optima.

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